The chromatic number of a graph of girth 5 on a fixed surface

We prove a color extension result implying that, for every fixed surface S, there are only finitely many 4-color-critical graphs of girth 5 on S. The result is best possible in the sense that there are infinitely many 4-color-critical graphs of girth 4 on S, except when S is the sphere. As a consequence, the chromatic number of graphs of girth 5 on S can be found in polynomial time.