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Unpolarized and polarized inelastic neutron scattering studies in single crystals of the triangular-lattice (TL) antiferromagnets TbInO₃ and TbIn_0.95Mn_0.05O₃ are reported. Broad gapless magnetic excitations are located at the TL Brillouin zone boundary. They show a weak enhancement near the M points at the lowest energies, and shift to the K points with increasing energy. At higher energies, a broad dispersing excitation branch, also centered at the zone boundary, is observed after a gap. No signs of magnetic order are found down to the temperatures 100 times smaller than the effective interaction energy given by the excitation bandwidth, indicating a very strong frustration. The fluctuating magnetic moment exceeds one half of the Tb³⁺ free-ion value and is confined to the TL plane. These observations strongly suggest a triangular-lattice-based spin liquid state in TbInO₃.