  We consider interpretable topological spaces and topological groups in a
$p$-adically closed field $K$. We identify a special class of "admissible
topologies" with topological tameness properties like generic continuity,
similar to the topology on definable subsets of $K^n$. We show every
interpretable set has at least one admissible topology, and every interpretable
group has a unique admissible group topology. We then consider definable
compactness (in the sense of Fornasiero) on interpretable groups. We show that
an interpretable group is definably compact if and only if it has finitely
satisfiable generics (fsg), generalizing an earlier result on definable groups.
As a consequence, we see that fsg is a definable property in definable families
of interpretable groups, and that any fsg interpretable group defined over
$\mathbb{Q}_p$ is definably isomorphic to a definable group.
