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We report single-crystal time-of-flight inelastic neutron scattering measurements of the magnetic excitations in the pseudospin-1/2 triangular-lattice near-Heisenberg antiferromagnet Ba₃CoSb₂O₉. We observe sharp magnons throughout reciprocal space with strongly renormalized dispersions and multiple soft modes compared to a linear spin-wave dispersion. At higher energies there is a relatively very strong and highly-structured continuum of excitations with overall weight and intensity modulations in energy and momentum that cannot be captured by a two magnon linear spin-wave cross-section. We attribute the strong dispersion renormalizations that lead to avoided magnon decays and large transfer of spectral weight from magnons to continuum states to the effect of quantum fluctuations and interactions beyond the spin wave approximation. Phys. Rev. B 102, 064421 (2020)