We generalize the Lipschitz constant to Whitney's functions and prove that any Whitney's function defined on a non-empty subset of $\bold R^n$ extends to a Whitney's function of domain $\bold R^n$ with the same constant. The proof uses an argument which refines the one used by Kirszbraun in the continuous case and, for this reason, holds only for $\bold R^n$ equiped with the euclidean norm. This constant is exactly the Lipschitz constant of the gradient of the extension and, therefore, this extension is minimal. We continue the paper with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case.