In this paper, the problem of distributed caching in dense wireless small cell networks (SCNs) is studied using mean field games (MFGs). In the considered SCN, small base stations (SBSs) are equipped with data storage units and cooperate to serve users' requests either from files cached in the storage or directly from the capacity-limited backhaul. The aim of the SBSs is to define a caching policy that reduces the load on the capacity-limited backhaul links. This cache control problem is formulated as a stochastic differential game (SDG). In this game, each SBS takes into consideration the storage state of the other SBSs to decide on the fraction of content it should cache. To solve this problem, the formulated SDG is reduced to an MFG by considering an ultra-dense network of SBSs in which the existence and uniqueness of the mean-field equilibrium is shown to be guaranteed. Simulation results show that this framework allows an efficient use of the available storage space at the SBSs while properly tracking the files' popularity. The results also show that, compared to a baseline model in which SBSs are not aware of the instantaneous system state, the proposed framework increases the number of served files from the SBSs by more than 69%.