Defining integer-valued functions in rings of continuous definable functions over a topological field

Let K be an expansion of either an ordered field (K, <=), or a valued field (K,v). Given a definable set X subset of Km let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P-minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension >= 2, then C(X) defines the subring Z. If K is P-minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f is an element of C(X) which take values in Z is definable in C(X).