  We study stability of metric approximations of countable groups with respect
to groups endowed with ultrametrics, the main case study being a $p$-adic
analogue of Ulam stability, where we take $\mathrm{GL}_n(\mathbb{Z}_p)$ as
approximating groups instead of $\mathrm{U}(n)$. For finitely presented groups,
the ultrametric nature implies equivalence of the pointwise and uniform
stability problems, and the profinite one implies that the corresponding
approximation property is equivalent to residual finiteness. Moreover, a group
is uniformly stable if and only if its largest residually finite quotient is.
We provide several examples of uniformly stable groups, including finite
groups, virtually free groups, some groups acting on rooted trees, and certain
lamplighter and (Generalized) Baumslag--Solitar groups. We construct a finitely
generated group that is not uniformly stable. Finally, we prove and apply a
(bounded) cohomological criterion for stability of a finitely presented group.

Group Theory