  Using thermodynamic measurements, neutron diffraction, nuclear magnetic
resonance, and muon spin relaxation, we establish putative quantum spin-liquid
behavior in Ba$_3$InIr$_2$O$_9$, where unpaired electrons are localized on
mixed-valence Ir$_2$O$_9$ 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.
