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

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