  We prove generic differentiability in $P$-minimal theories, strengthening an
earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and
Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups.
Specifically, let $G$ be an $n$-dimensional definable group in a highly
saturated model $M$ of a $P$-minimal theory. Then there is an open definable
subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$,
and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally,
the generic differentiability theorem immediately implies a classification of
interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and
Peterzil.
