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Similarly to warm dark matter which features a cut-off in the matter power spectrum due to free-streaming, many interacting dark matter models predict a suppression of the matter power spectrum on small length scales through collisional damping. Forecasts for 21cm line intensity mapping have shown that an instrument like the SKA will be able to probe a suppression of power in warm dark matter scenarios in a statistically significant way. Here we investigate the implications of these findings on interacting dark matter scenarios, particularly dark matter-neutrino interactions, which we use as an example. Using a suite of cosmological $N$-body simulations, we demonstrate that interacting scenarios show a suppression of the non-linear power spectrum similar to warm dark matter models. This implies that 21cm line intensity mapping will be able to set the strongest limits yet on dark matter-neutrino scattering, improving the constraints by two orders of magnitude over current Lyman-$α$ bounds, and by four orders of magnitude over cosmic microwave background and baryon acoustic oscillations limits. However, to distinguish between warm dark matter and interacting scenarios, our simulations show that percent-level precision measurements of the matter power spectrum at redshifts $z\gtrsim15$ are necessary, as the key features of interacting scenarios are washed out by non-linear evolution at later times.