In [12], P. Scowcroft and L. van den Dries proved a cell decomposition theorem for p-adically closed fields. We work here with the notion of P-minimal fields defined by D. Haskell and D. Macpherson in [6]. We prove that a P-minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [8].

In this paper, we prove a definable version of Kirszbraun's theorem in a non‐Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function, where and, that is λ‐Lipschitz in the first variable, extends to a definable function that is λ‐Lipschitz in the first variable.