A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ 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.

Given a non-oscillating gradient trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ of a real analytic function $f$, we show that the limit $v$ of the secants at the limit point $0$ of $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ along the trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ is an eigenvector of the limit of the direction of the Hessian matrix Hess$\left( f \right)$ at $0$ along $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.

Let $M\subset \mathbb{R}^n$ be a connected component of an algebraic set $\varphi^{-1}(0)$, where $\varphi$ is a polynomial of degree $d$. Assume that $M$ is contained in a ball of radius $r$. We prove that the geodesic diameter of $M$ is bounded by $2r\nu(n)d(4d-5)^{n-2}$, where $\nu(n)=2{\Gamma({1}/{2})\Gamma(({n+1})/{2})}{\Gamma({n}/{2})}^{-1}$. This estimate is based on the bound $r\nu(n)d(4d-5)^{n-2}$ for the length of the gradient trajectories of a linear projection restricted to $M$.