Summary: Kirszbraun’s Theorem states that every Lipschitz map \(S \rightarrow \mathbb R^n\), where \(S\subseteq \mathbb R^m\), has an extension to a Lipschitz map \(\mathbb R^m \rightarrow \mathbb R^n\) with the same Lipschitz constant. Its proof relies on Helly’s Theorem: every family of compact subsets of \(\mathbb R^n\), having the property that each of its subfamilies consisting of at most \((n + 1)\) sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.