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Recent experiments suggest that the superconducting order parameter of Sr₂RuO₄ has two components. A two-component order parameter has multiple degrees of freedom that can result in low-energy collective modes or the formation of domain walls, the dynamics of which can show up in sound attenuation experiments. We measured ultrasound attenuation across the superconducting transition of Sr₂RuO₄ and find that the attenuation for compressional sound increases by a factor of seven immediately below T_c, in sharp contrast with what is found in both conventional (s-wave) and high-T_c (d-wave) superconductors. Our observations appear to be most consistent with the presence of domain walls between different configurations of the superconducting state. The formation of domains naturally explains observations such as the smallness of time reversal symmetry breaking signal at T_c, and telegraph noise in critical current experiments. The fact that we observe an increase in sound attenuation only for compressional strains further suggests an inhomogeneous superconducting state formed of two distinct, accidentally-degenerate superconducting order parameters that are not symmetry-related.