  Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal
structure is not definably compact, then $G$ contains a definable torsion-free
subgroup of dimension one. We prove here a $p$-adic analogue of the
Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be
an abelian group definable in a $p$-adically closed field $M$. If $G$ is not
definably compact then there is a definable subgroup $H$ of dimension one which
is not definably compact. In a future paper we will generalize this to
non-abelian $G$.
