  We study infinite groups interpretable in three families of valued fields:
$V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show
that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal
then it is abelian-by-finite.
  Along the way, we associate with any infinite interpretable group an infinite
type-definable subgroup which is definably isomorphic to a group in one of four
distinguished sorts: the underlying valued field $K$, its residue field
$\mathbf{k}$ (when infinite), its value group $\Gamma$, or $K/\mathcal{O}$,
where $\mathcal{O}$ is the valuation ring.
  Our work uses and extends techniques developed in [11] to circumvent
elimination of imaginaries.
