  We generalize two of our previous results on abelian definable groups in
$p$-adically closed fields to the non-abelian case. First, we show that if $G$
is a definable group that is not definably compact, then $G$ has a
one-dimensional definable subgroup which is not definably compact. This is a
$p$-adic analogue of the Peterzil-Steinhorn theorem for o-minimal theories.
Second, we show that if $G$ is a group definable over the standard model
$\mathbb{Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable
groups over $\mathbb{Q}_p$ are open subgroups of algebraic groups, up to finite
factors. We also prove that $G^0 = G^{00}$ when $G$ is a definable subgroup of
a linear algebraic group, over any model.
