Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions

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.