This work reports a novel plasma wave observation in the near-Sun solar wind: frequency-dispersed ion acoustic waves. Similar waves have previously been reported in association with interplanetary shocks or planetary bow shocks, but the waves reported here occur throughout the solar wind sunward of ∼60 solar radii, far from any identified shocks. The waves reported here vary their central frequency by factors of 3–10 over tens of milliseconds, with frequencies that move up or down in time. Using a semiautomated identification algorithm, thousands of wave instances are recorded during each near-Sun orbit of the Parker Solar Probe spacecraft. Wave statistical properties are determined and used to estimate their plasma frame frequency and the energies of protons most likely to be resonant with these waves. Proton velocity distribution functions are explored for one wave interval, and proton enhancements that may be consistent with proton beams are observed. A conclusion from this analysis is that properties of the observed frequency-dispersed ion acoustic waves are consistent with driving by cold, impulsively accelerated proton beams near the ambient proton thermal speed. Based on the large number of observed waves and their properties, it is likely that the impulsive proton beam acceleration mechanism generating these waves is active throughout the inner heliosphere. This may have implications for the acceleration of the solar wind.