  We prove that a (globally) subanalytic p-adic function which is locally
Lipschitz continuous with some constant C is piecewise (globally on each piece)
Lipschitz continuous with possibly some other constant, where the pieces can be
taken subanalytic. We also prove the analogous result for a subanalytic family
of functions depending on p-adic parameters. The statements also hold in a
semi-algebraic set-up and also in finite extensions of the field of p-adic
numbers. These results are p-adic analogues of results of K. Kurdyka over the
real numbers. To encompass the total disconnectedness of p-adic fields, we need
to introduce new methods adapted to the p-adic situation.
