[The following is a vast simplification of an answer I gave a bit earlier, so I deleted most of the earlier answer. For anyone who cares about the sanctity of upvoting points it is all OK since I am certain anyone who upvoted the earlier answer would upvote this one too.]

Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are nonzero abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective. In what follows one can work throughout with elliptic curves if that is preferred, and the main idea can be carried out in more elementary terms for elliptic curves, but I give the general case because it is more attractive.

The first thing to observe is that if $n$ is a nonzero integer then its effect on the nonzero semi-abelian identity components of the special fibers of the Neron models is nonzero (by viewing such identity components as extensions of abelian varieties by tori, and noting that a nonzero integer acts in a nonzero way on any nonzero torus and on any nonzero abelian variety).

The next thing to observe is the consequence that an isogeny of abelian varieties induces an isogeny between such identity components of special fibers is an isogeny of abelian varieties admits an "isogeny inverse": a map in the opposite direction such that the composition in both directions is multiplication by a common nonzero integer. In particular, an isogeny between abelian varieties over $F$ with semistable reduction induces an isogeny between identity components of special fibers of the Neron model. (This already settles the case of elliptic curves!)

Consequently, for our purpose we can replace either of $A$ or $B$ with an $F$-isogenous abelian variety (which always has semistable reduction too: this is well-known for elliptic curves, and follows from Grothendieck's inertial semistable reduction criterion in general). Thus, in case either of $A$ or $B$ are not $F$-simple (though they are for the case of elliptic curves) we can apply the Poincare reducibility theorem to reduce to the case when both are $F$-simple. But then any nonzero $F$-homomorphism between them is an isogeny and hence as we have seen above induces an isogeny between identity components of special fibers of the Neron models. In particular, the induced map between such special fibers is nonzero.

[The following is a vast simplification of an answer I gave a bit earlier, so I deleted most of the earlier answer. For anyone who cares about the sanctity of upvoting points it is all OK since I am certain anyone who upvoted the earlier answer would upvote this one too.]

Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective. In what follows one can work throughout with elliptic curves if that is preferred, and the main idea can be carried out in more elementary terms for elliptic curves, but I give the general case because it is more attractive.

The first thing to observe is that if $n$ is a nonzero integer then its effect on the semi-abelian identity components of the special fibers of the Neron models is an isogeny (by viewing such identity components as extensions of abelian varieties by tori, and noting that a nonzero integer acts as an isogeny on any torus and on any abelian variety).

The next thing to observe is since an isogeny of abelian varieties admits a map in the opposite direction such that the composition in both directions is multiplication by a common nonzero integer, the preceding observation about the effect of nonzero integers on the identity components of the special fibers implies that an isogeny between abelian varieties over $F$ with semistable reduction induces an isogeny between identity components of special fibers of the Neron model. (This already settles the case of elliptic curves in the question posed above.)

Consequently, for our purpose we can replace either of $A$ or $B$ with an $F$-isogenous abelian variety (which always has semistable reduction too: this is well-known for elliptic curves, and follows from Grothendieck's inertial semistable reduction criterion in general). Thus, in case either of $A$ or $B$ are not $F$-simple (though they are $F$-simple for the case of elliptic curves) we can apply the Poincare reducibility theorem to reduce to the case when both are $F$-simple. But then any nonzero $F$-homomorphism between them is an isogeny and hence as we have seen above induces an isogeny between identity components of special fibers of the Neron models. In particular, the induced map between such special fibers is nonzero.

The real work is in the case of bad reduction, as otherwise one can use finite etale torsion levels of the entire elliptic curve (the crux being that passage to the special fiber is an equivalence of categories between finite etale schemes over a henselian local ring and over its residue field, so in particular reduction is a faithful functor on such schemes). Below we give a method that treats all cases of semistable reduction in all dimensions by a uniform method, in fact ultimately using the same "crux" mentioned above (with help from Zariski's Main Theorem to get around the headache of quasi-finite maps that may not be finite).

Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective.

We may and do apply scalar extension on $R$ so that it is complete (or just henselian is good enough for what follows, if one wants to be more "algebraic"). We may also assume $A \ne 0$, and that $A$ is $F$-simple (since in the case of semistable reduction isogenies between abelian varieties induce isogenies between identity components of special fibers of Neron models, as we see by factoring an isogeny through multiplication by some nonzero integer).

Pick a prime $\ell \ne {\rm{char}}(k)$ and consider the $\ell$-power torsion $\mathcal{A}^0[\ell^n]$, where $\mathcal{A}^0$ is the open "relative identity component" (i.e., the open complement in $\mathcal{A}$ of the closed locus of non-identity components in the closed fiber). The hypothesis of semistable reduction implies (by fibral considerations, due to flatness of the Neron model over $R$) that these are quasi-finite etale separated commutative $R$-groups with special fiber $\mathcal{A}^0_k[\ell^n]$ that has order at least $\ell^n$ (recall we arranged $A \ne 0$).

By Zariski's Main Theorem, any quasi-finite etale separated scheme $X$ over a henselian local ring $C$ is uniquely a disjoint union $X_{\rm{f}} \coprod X'$ of a finite $C$-scheme $X_{\rm{f}}$ and a $C$-scheme $X'$ with empty special fiber. The formation of $X_{\rm{f}}$ is functorial in $X$ and compatible with direct products, so if $X$ is a $C$-group then $X_{\rm{f}}$ is an open and closed $R$-subgroup.

We conclude that $\mathcal{A}^0[\ell^n]_{\rm{f}}$ is a finite etale $R$-group with order at least $\ell^n$. For finite etale schemes over a henselian local ring $C$, passage to the special fiber sets up an equivalence of categories (a very familiar fact for $C$ a complete discrete valuation ring with perfect residue field, where it amounts to the Galois theory of the residue field encoding the structure of unramified extensions of the fraction field). Thus, the natural map $${\rm{Hom}}_R(\mathcal{A}^0[\ell^n]_{\rm{f}}, \mathcal{B}^0[\ell^n]_{\rm{f}}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k[\ell^n],\mathcal{B}^0_k[\ell^n])$$ is an equality! In particular, if $h:A \rightarrow B$ is an $F$-homomorphism whose induced map $\mathcal{A}_k^0 \rightarrow \mathcal{B}_k^0$ vanishes then the map between Neron models vanishes on $\mathcal{A}^0[\ell^n]_{\rm{f}}$, and hence $h$ vanishes on the generic fiber $(\mathcal{A}^0[\ell^n]_{\rm{f}})_F$ for all $n \ge 1$. This generic fiber has order at least $\ell^n$, so $\ker h$ contains finite etale $F$-subgroups of order at least $\ell^n$ for all $n \ge 1$.

To conclude that $\ker h = A$, it now suffices to show that if $A$ is an $F$-simple abelian variety over any field $F$ and $H$ is a closed $F$-subgroup scheme of $A$ containing finite etale $F$-subgroups $H_n$ with unbounded order as $n$ varies then $H=A$. Consider the subgroup of $H(F_s)$ generated by the groups $H_n(F_s)$. This is a ${\rm{Gal}}(F_s/F)$-stable subgroup of $H(F_s)$, so the Zariski closure in $H_{F_s}$ of that subgroup is a closed $F_s$-subgroup scheme $Z'$ of $H_{F_s}$ that is preserved by the natural $F_s/F$-descent datum on $H_{F_s}$ encoding its $F$-descent $H$ and is geometrically reduced over $F_s$, hence is $F_s$-smooth. Thus, by Galois descent $Z'$ descends to an $F$-smooth closed $F$-subgroup $Z \subset H$ that contains every $H_n$ by design. For the abelian variety $Z^0 \subset A$, we have that $Z/Z^0$ is finite, so the orders of the $F$-subgroups $Z^0 \cap H_n$ are unbounded as $n$ varies. This implies $Z^0 \ne 0$. But $A$ is $F$-simple by hypothesis (!), so $Z^0 = A$. By design we have $Z^0 \subset H$, so $H=A$ as desired.

[The following is a vast simplification of an answer I gave a bit earlier, so I deleted most of the earlier answer. For anyone who cares about the sanctity of upvoting points it is all OK since I am certain anyone who upvoted the earlier answer would upvote this one too.]

Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are nonzero abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective. In what follows one can work throughout with elliptic curves if that is preferred, and the main idea can be carried out in more elementary terms for elliptic curves, but I give the general case because it is more attractive.

The first thing to observe is that if $n$ is a nonzero integer then its effect on the nonzero semi-abelian identity components of the special fibers of the Neron models is nonzero (by viewing such identity components as extensions of abelian varieties by tori, and noting that a nonzero integer acts in a nonzero way on any nonzero torus and on any nonzero abelian variety).

The next thing to observe is the consequence that an isogeny of abelian varieties induces an isogeny between such identity components of special fibers is an isogeny of abelian varieties admits an "isogeny inverse": a map in the opposite direction such that the composition in both directions is multiplication by a common nonzero integer. In particular, an isogeny between abelian varieties over $F$ with semistable reduction induces an isogeny between identity components of special fibers of the Neron model. (This already settles the case of elliptic curves!)

Consequently, for our purpose we can replace either of $A$ or $B$ with an $F$-isogenous abelian variety (which always has semistable reduction too: this is well-known for elliptic curves, and follows from Grothendieck's inertial semistable reduction criterion in general). Thus, in case either of $A$ or $B$ are not $F$-simple (though they are for the case of elliptic curves) we can apply the Poincare reducibility theorem to reduce to the case when both are $F$-simple. But then any nonzero $F$-homomorphism between them is an isogeny and hence as we have seen above induces an isogeny between identity components of special fibers of the Neron models. In particular, the induced map between such special fibers is nonzero.