In the setting of L-convex spaces, this paper proves a result ensuring the existence of pure strategy Nash equilibrium for non-cooperative games with infinite countable players. Following the method introduced by Nikaido and Isoda (1955), this paper defines an aggregate payoff function, and next then, determines some restrictions on the aggregate function that guarantee the existence of pure strategy Nash equilibrium for non-cooperative games with infinite countable players. In the process of proof, the method adopted by this paper is to apply the continuous unity partition theorem and a famous fixed point theorem due to Gorniewicz (1975). Finally, some new perturbed saddle point theorems are obtained. Our results generalize and improve the known Nash equilibrium existence results for non-cooperative games with finite players in the literature. Our results improve and unify the corresponding results in the recently existing literatures.