Peterzil and Steinhorn proved that if $G$ is a definable group in an o-minimal structure, and $G$ is not definably compact, then there is a 1-dimensional definable subgroup of $G$ that is not definably compact. In this talk, we will consider the analogous statement for groups definable in $p$-adically closed fields. We will sketch a proof in the abelian case, and suggest a plan of attack for the non-abelian case. The proof of the Peterzil-Steinhorn theorem relies on connectedness in a critical step, and some new techniques are needed to adapt their proof to the totally disconnected $p$-adic setting. We will also discuss potential generalizations to groups definable in dp-minimal fields. This is joint work with Ningyuan Yao.