Let F be a non-principal ultraproduct of finite fields, T its theory, and A a subfield of F which is relatively algebraically closed in F. It is known that any two models of T which contain A as a relatively algebraically closed subfield are elementarily equivalent over A. Theorem: If A is not pseudo-finite, then T has no prime model over A. The proof of this result is fairly easy when A is countable: one just uses that prime models over A are atomic. The proof when A is uncountable is more involved, as it involves constructing 2^{|A|} non-isomorphic models of T which are of transcendence degree 1 over A. We also discuss the existence or non-existence of \kappa-prime models of T (i.e., \kappa-saturated models of T containing A and which A-embed into any \kappa-saturated model of T containing A), for regular uncountable \kappa such that \kappa =\kappa^{<\kappa}.