We study the behavior of canonical height functions $h{̂}_f$, associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $h{̂}_f$ on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a...