Unique range sets in positive characteristic

We give examples and non-examples of unique range sets for ultrametric entire functions in positive characteristic, in an algebraically closed complete ultrametric field. In particular, if n greater than or equal to 4, we show there exists a unique range set with n elements for the family of non-constant one variable polynomials in any characteristic. As far as a pure existence theorem goes, this is the best one can hope for, as there are no 3 point unique range sets in characteristic three. For all prime powers q = p(n) greater than or equal to 3, we construct an affinely rigid set of q elements which is not a unique range set for the non-constant one variable polynomials in characteristic p. All our results hold equally well for the family of non-constant one variable non-Archimedean entire functions.