Caching popular contents at the edge of the network can positively impact the performance of wireless systems and increase their future sustainability. While the benefits of caching are now rather evident at network level, very few works have focused on its positive impact on the physical layer. This paper aims at showing that significant performance enhancements can indeed be experienced at the physical layer as well, especially in terms of network interference footprint. To this end, we consider a full-duplex (FD) small-cell (SC) network consisting of non-cooperative base stations with caching capabilities, which communicate simultaneously with both downlink users and wireless backhaul nodes. We first propose a novel caching model seeking to mimic a geographical caching policy based on local files popularity and calculate the corresponding cache hit probability. Then, using tools from stochastic geometry, we study the probability of successfully transmitting a file in the considered network. Numerical results corroborate our theoretical findings and highlight remarkable performance gains when moving from cache-free to cache-aided FD SC networks.