Lipschitz extensions of definable p-adic functions

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 $f : X \times Y \to $\mathbb{Q}$_p^s$,where$X\subset $\mathbb{Q}$_p$and$Y \subset $\mathbb{Q}$_p^r$, that is $\lambda$-Lipschitz in the first variable, extends to a definable function $\tilde{f}:{\mathbb{Q}}_p\times Y\to {\mathbb{Q}}_p^s$ that is $\lambda$-Lipschitz in the first variable.