2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165643For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of such maps is derived in some cases. Besides, these results are used to characterize the separating bijections between scalar-valued Lipschitz function spaces when $Y$ is compact.17 pages; no figuresFunctional Analysis47B38 (Primary); 46E40, 46H40, 47B33 (Secondary)text