Reconfigurable intelligent surfaces (RISs) bring various benefits to the current and upcoming wireless networks, including enhanced spectrum and energy efficiency, soft handover, transmission reliability, and even localization accuracy. These remarkable improvements result from the reconfigurability, programmability, and adaptation capabilities of RISs for fine-tuning radio propagation environments, which can be realized in a cost- and energy-efficient manner. In this paper, we focus on the upgrade of the existing fifth-generation (5G) cellular network with the introduction of an RIS owning a full-dimensional uniform planar array structure for unleashing advanced three-dimensional connectivity. The deployed RIS is exploited for serving unmanned aerial vehicles (UAVs) flying in the sky with ultra-high data rate, a challenging task to be achieved with conventional base stations (BSs) that are designed mainly to serve ground users. By taking into account the line-of-sight probability for the RIS-UAV and BS-UAV links, we formulate the average achievable rate, analyze the effect of environmental parameters, and make insightful performance comparisons. Simulation results show that the deployment of RISs can bring impressive gains and significantly outperform conventional RIS-free 5G networks.