We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3Q2 that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake projective plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.
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