We propose the combination of subwavelength, two-dimensional atomic arrays and Rydberg interactions as a powerful platform to realize strong, coherent interactions between individual photons with high fidelity. The atomic spatial ordering guarantees efficient atom-light interactions without the possibility of scattering light into unwanted directions, allowing the array to act as a perfect mirror for individual photons. In turn, Rydberg interactions enable single photons to alter the optical response of the array within a potentially large blockade radius Rb , which can effectively punch a large "hole" for subsequent photons. We show that such a system enables a coherent photon-photon gate or switch, with a significantly better error scaling (∼Rb-4 ) than in a disordered ensemble. We also investigate the optical properties of the system in the limit of strong input intensities and show that this many-body quantum driven dissipative system can be modeled well by a semiclassical model based on holes punched in a classical mirror.