We perform a series of cosmological hydrodynamic simulations to investigate the effects of non-gravitational dark matter (DM) interactions on the intergalactic medium (IGM). In particular, we use the ETHOS framework to compare statistics of the Lyman-α forest in cold dark matter (CDM) with an alternative model in which the DM couples strongly with a relativistic species in the early universe. These models are characterized by a cut-off in the linear power spectrum, followed by a series of ‘dark acoustic oscillations’ (DAOs) on sub-dwarf scales. While the primordial cut-off delays the formation of the first galaxies, structure builds up more rapidly in the interacting DM model compared to CDM. We show that although DAOs are quickly washed away in the non-linear clustering of DM at z ≲ 10, their signature can be imprinted prominently in the Lyman-α flux power spectrum at z > 5. On scales larger than the cut-off (k ∼ 0.08 s km−1 for the specific model considered here), the relative difference to CDM is reminiscent of a warm dark matter (WDM) model with a similar initial cut-off; however, the redshift evolution on smaller scales is distinctly different. The appearance and disappearance of DAOs in the Lyman-α flux spectrum provides a powerful way to distinguish interacting DM models from WDM and, indeed, variations in the thermal history of the IGM that may also induce a small-scale cut-off.