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.