In this paper we investigate stochastic partial differential equations with jumps in infinite dimensions. The key motivation of this paper is, under a local Lipschitz condition but without a linear growth condition, to give an existence-and-uniqueness theorem (Khasminskii-type theorem), where the classical existence-and-uniqueness result can be regarded as a special case, and then to discuss the almost sure asymptotic stability of the solutions. Moreover, as a by-product, we also derive that the solutions are weakly attracted. Finally, several examples are constructed to demonstrate our results.