Specializing p-torsion in a family of elliptic surfaces Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$. Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective morphism of relative dimension 2. Let $f:Y\longrightarrow \mathbb{P}_R^1$ be a morphism admitting  a smooth section, such that $f_K: Y_K\longrightarrow\mathbb{P}_K^1$ and $f_k: Y_k\longrightarrow\mathbb{P}_k^1$ are elliptic surfaces of Kodaira dimension one, whose only singular fibers are of type $I_m$ ($m$ is allowed to vary across the singular fibers). Is there an example of $Y$ as above so that, furthermore, the specialization map
$$\operatorname{Pic}^\tau(Y_K/\mathbb{P}^1_K)[p]\longrightarrow \operatorname{Pic}^\tau(Y_k/\mathbb{P}^1_k)[p]$$
is not injective?
 A: As Will Sawin points out, there is no $p$-torsion in the Picard in the first place. Here is my attempt of an answer.
Let $f_{\bar{K}}:Y_{\bar{K}}\longrightarrow\mathbb{P}_{\bar{K}}^1$ be the restriction of $f$ over the geometric generic point of $\mathrm{Spec}R$. By the Lefschetz principle, we may assume that everything is defined over $\mathbb{C}$. Let $S\subset Y_{\bar{K}}$ be the smooth section of $f_{\bar{K}}$ and suppose $L\in\mathrm{Pic}(Y_{\bar{K}}/\mathbb{P}_{\bar{K}}^1)[p]$ is a nontrivial line bundle. Then $L^p=f^\ast\mathcal{O}_{\mathbb{P}^1_{\bar{K}}}(n)$ for some $n\in \mathbb{Z}$. Intersecting with $S$ we have $n\in p\mathbb{Z}$. Replacing $L$ with $L\otimes f^\ast\mathcal{O}_{\mathbb{P}^1_{\bar{K}}}(-n/p)$ we have a nontrivial $L\in\mathrm{Pic}(Y_{\bar{K}})[p]$, thus a nontrivial étale cover. On the other hand, since $\kappa(X_{\bar{K}})=1$, we have $\chi(\mathcal{O}_{X_\bar{K}})>0$, hence $\pi_1(Y_{\bar{K}})\cong\pi_1^{\mathrm{orb}}(\mathbb{P}^1_{\bar{K}})$ by Theorem 2.3 in Chapter II Section 2 of this book. As there are no multiple fibers, we have $\pi_1^{\mathrm{orb}}(\mathbb{P}^1_{\bar{K}})=0$, a contradiction.
