The starting point of this question is the (presumably) well-known theorem (the proof I know is from *Abelian $\ell$-adic representations and elliptic curves* from J-P.Serre in which it is a lemma for $n=2$ and an exercise for $n>2$; which suggests that the result was already classical in the 60s).

**Theorem I:** If $G$ is a closed subgroup of $\operatorname{SL}_{n}(\mathbb Z_{p})$ which surjects on $\operatorname{SL}_{n}(\mathbb F_{p})$ and if $p≥5$, then $G=\operatorname{SL}_{n}(\mathbb Z_{p})$.

The theorem is optimal with respect to all hypotheses in the sense that there exists proper subgroups of $\operatorname{SL}_{2}(\mathbb Z_{p})$ mapping onto $\operatorname{SL}_{2}(\mathbb F_{p})$ when $p=2,3$ (a fact that played a role in the original proof of the modularity of semi-stable elliptic curves by A.Wiles, if I am not mistaken) and in the sense that for all $n≥2$ and all prime $p$, there exist discrete valuation rings $A$ of mixed characteristic $(0,p)$ such that $\operatorname{SL}_{n}(A)$ contains proper subgroups mapping onto $\operatorname{SL}_{n}(A/\mathfrak m)$ (just take a completely ramified $A$ over $\mathbb Z_{p}$ and consider $\operatorname{SL}_{2}(\mathbb Z_{p})$ inside $\operatorname{SL}_{2}(A)$).

A slightly less well-known fact is that the theorem admits the following generalization, due to N.Boston.

**Theorem II:** Let $A$ be a complete local noetherian ring with finite residual characteristic $p\neq 2$. If $G$ is a closed subgroup of $\operatorname{SL}_{n}(A)$ which surjects on $\operatorname{SL}_{n}(A/\mathfrak m^2)$, then $G=\operatorname{SL}_{n}(A)$.

This is in the appendix of *On p-adic analytic families of Galois representations*.
Compositio Math. 59 (1986), no. 2, 231–264 by B.Mazur and A.Wiles and again, this theorem is optimal in the sense that there exists a proper subgroup of $\operatorname{SL}_{2}(\mathbb Z_{2})$ surjecting on $\operatorname{SL}_{2}(\mathbb Z/4\mathbb Z)$.

Now my actual question. Let $A$ be a complete local noetherian ring (UPDATE: domain, actually) of mixed characteristic $(0,p)$ with $p≠2$ UPDATE: which one can assume to be a discrete valuation ring if necessary. Suppose $G$ is a closed subgroup of $\operatorname{SL}_{n}(A)$ which surjects on $\operatorname{SL}_{n}(A/\mathfrak m)$.

Among the pre-images in $G$ of non-identity unipotents elements in $\operatorname{SL}_{n}(A/\mathfrak m)$, is it true that there exists a unipotent element (that is to say an element $\sigma$ such that $\sigma-1$ is nilpotent)?

Granted theorem II, this is obviously true if $G$ maps onto $\operatorname{SL}_{n}(A/\mathfrak m^2)$. Without this hypothesis, it looks dubious to me but nevertheless, the obvious counterexamples to theorem I coming from ramified rings do not provide counterexamples to this claim and I confess that I don't quite know how to construct other counter-examples. Taking this into account, another more general point of view on the question would be the following.

What are the subgroups of $\operatorname{SL}_{n}(A)$ which do not map onto $\operatorname{SL}_{n}(A/\mathfrak m^2)$ but which do map onto $\operatorname{SL}_{n}(A/\mathfrak m)$?

Perhaps group cohomology of $\operatorname{SL}_{n}$ would help then. Any positive result, even in the case $n=2$ and $p>3$ would already be of interest to me.

UPDATE: Jim Humphreys asks in comments what I mean by a unipotent element in a matrix group. I meant an element $\sigma$ such that $\sigma-1$ is nilpotent, but I realize now that this might not be so great a definition when the ring of coefficients is not a domain, or at least reduced. So perhaps it is better to take $A$ a domain in the above, or even a discrete valuation ring. Also, even though Tim Dokchister answered the first question in the negative, I would be interested to know if someone has something to say for infinite closed subgroup $G$.

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