 Nonstandard prime in an elementary extension of the integers

Name every integer and add a new constant asserted to be prime and unequal to every named integer. Every finite part is satisfied by an ordinary prime outside a finite list, using Euclid's theorem. Compactness theorem with the elementary diagram gives an elementary extension with the desired prime. A countable elementary substructure retaining that prime cannot be isomorphic to the integer ring, since any unital ring isomorphism fixes every standard integer.