  We study infinite groups interpretable in power bounded $T$-convex,
$V$-minimal or $p$-adically closed fields. We show that if $G$ is an
interpretable definably semisimple group (i.e., has no definable infinite
normal abelian subgroups) then, up to a finite index subgroup, it is definably
isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and
$G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying
the interaction of $G$ with four distinguished sorts: the valued field $K$, the
residue field $\mathbf{k}$, the value group $\Gamma$, and the closed $0$-balls
$K/\mathcal{O}$.
