1) A ball (of finite radius) is compact, and a punctured ball is not, so they cannot be diffeomorphic.
2) A theorem of Serre (Topology, 1968Topology, 1965, vol. 3, p. 409-412) classifies all compact $p$-adic manifolds. The result is:
a) Any such (non-empty) manifold is isomorphic to a finite disjoint union of balls. b) For positive integers $a$ and $b$, the union of $a$ balls is isomorphic to the union of $b$ ones if and only if $a\equiv b\pmod{p-1}$.
For example, the unit ball is isomorphic to the union of $p$ disjoint balls (the residue classes).
[Edit, 6/15/2012: I corrected and added the reference to Serre's paper.]
