In this paper we investigate nonzero-sum games for continuous-time jump processes with Borel state spaces. The optimality criterion to be considered is the expected average payoff criterion. The action spaces for the players are Borel spaces, and the reward and transition rates can be possibly unbounded. Under suitable conditions, we introduce the auxiliary static games and prove the existence of a stationary Nash equilibrium in the class of all randomized history-dependent strategy profiles via a technique of discounted approximation. Moreover, an example is given to illustrate the optimality conditions.