We present the first detailed analysis of the effects of dissipative dark matter on stellar streams. As a concrete example, we generate a cosmological hydrodynamic zoom-in simulation of a Milky Way-mass galaxy, assuming that the dark matter consists of Cold Dark Matter (CDM) with a sub-component ($\sim6\%$) of Atomic Dark Matter (ADM). The ADM subcomponent behaves as collisional, efficiently dissipative gas and allows for the formation of dense compact objects that enhance the central density of satellite galaxies, making them more resistant to tidal disruption. We show that stellar streams with stellar mass $M_{\rm{tot}, \star} \gtrsim 10^{5.5} \ \text{M}_\odot$ form later and exhibit prolonged star formation throughout their evolution, as compared to their CDM counterparts. Changes to star formation history are reflected on the chemical tracks of the stellar stream stars, where the youngest have enhanced [Fe/H] and [Mg/Fe] in the presence of ADM. Furthermore, a population of low-mass satellites with high ADM mass fractions is identified at low pericenter distances, which may affect the population of streams at $M_{\rm{tot}, \star} \lesssim 10^{5.5} \ \text{M}_\odot$. The results of this study should generalize to other dark matter models that lead to inner-density enhancements in satellites, such as elastic self-interacting dark matter in the gravothermal collapse regime.