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Cache-aided content delivery is studied in a multi-server system with $P$ servers and $K$ users, each equipped with a local cache memory. In the delivery phase, each user connects randomly to any $ρ$ out of $P$ servers. Thanks to the availability of multiple servers, which model small-cell base stations (SBSs), demands can be satisfied with reduced storage capacity at each server and reduced delivery rate per server; however, this also leads to reduced multicasting opportunities compared to the single-server scenario. A joint storage and proactive caching scheme is proposed, which exploits coded storage across the servers, uncoded cache placement at the users, and coded delivery. The delivery \textit{latency} is studied for both \textit{successive} and \textit{parallel} transmissions from the servers. It is shown that, with successive transmissions the achievable average delivery latency is comparable to the one achieved in the single-server scenario, while the gap between the two depends on $ρ$, the available redundancy across the servers, and can be reduced by increasing the storage capacity at the SBSs. The optimality of the proposed scheme with uncoded cache placement and MDS-coded server storage is also proved for successive transmissions.