For the first question, at least for $p=5, n=2$ there is a counterexample.:

The group $\text{SL}_2({\mathbb F}_5)$ has a 2-dimensional symplectic representation with character in ${\mathbb Q}(\sqrt 5)$ and Schur index 2, so it is realizable over $A={\mathbb Z}_5(\zeta_5)$. So $\text{SL}_2({\mathbb F}_5)$ injects in $\text{SL}_2(A)$ and reduces onto $\text{SL}_2(A/{\mathfrak m})$. But it is just a finite group, so it has no unipotent elements.

I don't know whether it generalizes to higher $p$ and $n$ though, and whether there are examples with `interesting' infinite closed subgroups of $\text{SL}_2(A)$.

For the first question, at least for $p=5, n=2$ there is a counterexample.:

The group $\text{SL}_2({\mathbb F}_5)$ has a 2-dimensional symplectic representation with character in ${\mathbb Q}(\sqrt 5)$ and Schur index 2, so it is realizable over $A={\mathbb Z}_5(\zeta_5)$. So $\text{SL}_2({\mathbb F}_5)$ injects in $\text{SL}_2(A)$ and reduces onto $\text{SL}_2(A/{\mathfrak m})$. But it is just a finite group, so it has no non-identity unipotent elements.

I don't know whether it generalizes to higher $p$ and $n$ though, and whether there are examples with `interesting' infinite closed subgroups of $\text{SL}_2(A)$.