In this paper, we prove an extension of Mahler's theorem on interpolation series, a celebrated result of p-adic analysis. Mahler's original result states that a function from N to Z is uniformly continuous for the p-adic metric d_p if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from a free monoid A* to Z, where d_p is replaced by the pro-p metric, the profinite metric on A* defined by p-groups.