We investigate the impact of topology on the existence of impurity subgap states in a time-reversal-invariant superconductor with an extended -wave pairing and strong spin-orbit coupling. By simply tuning the chemical potential, we access three distinct phases: topologically trivial -wave, topologically nontrivial -wave, and nodal superconducting phase. For a single potential impurity, we find subgap impurity bound states in the topological phase, but notably no subgap states in the trivial phase. This is in sharp contrast with the expectation that there would be no subgap state in the presence of potential impurities in -wave superconductors. These subgap impurity states have always finite energies for any strength of the potential scattering and, subsequently, the superconducting gap in the topological -wave phase survives but is attenuated in the presence of finite disorder. By creating islands of potential impurities, we smoothly connect the single impurity results to topological edge states of impurity islands. On the other hand, magnetic impurities lead to the formation of Yu-Shiba-Rusinov states in both the trivial and topological phases, which even reach zero energy at certain scattering strengths. We thus propose that potential impurities can be a very valuable tool to detect time-reversal-invariant topological superconductivity.