Superconductivity is a macroscopic quantum phenomenon that requires electron pairs to delocalize over large distances. A long-standing question is whether superconductivity can exist even if the electrons' kinetic energy is completely quenched, as is the case in a flat band. This is fundamentally a nonperturbative problem, since the interaction energy scale is the only relevant energy scale, and hence it requires going beyond the traditional Bardeen-Cooper-Schrieffer theory of superconductivity, which is perturbative by nature. In this work, we study a two-dimensional model of an isolated narrow band at partial filling with local attractive interactions, using numerically exact quantum Monte Carlo calculations. We focus on the case where the flat bands are topologically nontrivial, and hence the single-particle wave functions that span these bands cannot be completely spatially localized. Our calculations unambiguously demonstrate that the ground state is a superconductor; strikingly, the critical temperature scales nearly linearly with the interaction strength. Above the superconducting transition temperature, we find a broad pseudogap regime that exhibits strong pairing fluctuations and a tendency towards electronic phase separation. Introducing a small nearest-neighbor attraction suppresses superconductivity entirely and drives the system to phase separate. We discuss the possible relevance of superconductivity in this unusual regime to the physics of flat band moiré materials. © 2020 American Physical Society.