A definable, p-adic analogue of Kirszbraun's Theorem on extensions of Lipschitz maps

A direct application of Zorn’s Lemma gives that every Lipschitz map $f:X\subset {\mathbb{Q}}_p^n\to {\mathbb{Q}}_p^{\mathrm{\ell }}$ has an extension to a Lipschitz map $\tilde{f}:{\mathbb{Q}}_p^n\to {\mathbb{Q}}_p^{\mathrm{\ell }}$. This is analogous, but more easy, to Kirszbraun’s Theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset $\mathbb{R}$^n\to $\mathbb{R}$^\ell$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s Theorem. In the present paper, we prove in the $p$-adic context that $\tilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, Lipschitz retractions of ${\mathbb{Q}}_p^n$ to the topological closure of $X$ when $X$ is definable.