This paper presents and analyzes a reconfigurable intelligent surface (RIS)-based high-altitude platform (HAP) network. Stochastic geometry is used to model the arbitrary locations of the HAPs and RISs as a homogenous Poisson point process. Considering that the links between the HAPs, RISs, and users are $\kappa$--$\mu$ faded, the coverage and ergodic capacity of the proposed system are expressed. The analytically derived performance measures are verified through Monte Carlo simulations. Significant improvements in system performance and the impact of system parameters are demonstrated in the results. Thus, the proposed system concept can improve connectivity and data offloading in smart cities and dense urban environments.