Using thermodynamic measurements, neutron diffraction, nuclear magnetic resonance, and muon spin relaxation, we establish putative quantum spin-liquid behavior in Ba₃InIr₂O₉, where unpaired electrons are localized on mixed-valence Ir₂O₉ dimers with Ir^4.5+ ions. Despite the antiferromagnetic Curie-Weiss temperature on the order of 10 K, neither long-range magnetic order nor spin freezing are observed down to at least 20 mK, such that spins are short-range correlated and dynamic over nearly three decades in temperature. Quadratic power-law behavior of both the spin-lattice relaxation rate and specific heat indicates the gapless nature of the ground state. We envisage that this exotic behavior may be related to an unprecedented combination of the triangular and buckled honeycomb geometries of nearest-neighbor exchange couplings in the mixed-valence setting.