Generic differentiability and P-minimal groups

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 ⊆ G such that H is compactly dominated by H/H⁰⁰, and H/H⁰⁰ is a p-adic Lie group of the expected dimension.