Deep observations of galaxies reveal faint extended stellar components (hereafter ESCs) of streams, shells, and halos. These are a natural prediction of hierarchical galaxy formation, as accreted satellite galaxies are tidally disrupted by their host. We investigate whether or not global properties of the ESC could be used to test of dark matter, reasoning that they should be sensitive to the abundance of low-mass satellites, and therefore the underlying dark matter model. Using cosmological simulations of galaxy formation in the favoured Cold Dark Matter (CDM) and Warm Dark Matter (WDM) models ($m_{rm WDM}$=0.5,1,2 keV/$c^2$), which suppress the abundance of low-mass satellites, we find that the kinematics and orbital structure of the ESC is consistent across models. However, we find striking differences in its spatial structure, as anticipated -- a factor of $sim$10 drop in spherically averaged mass density between $sim$10% and $sim$75% of the virial radius in the more extreme WDM runs ($m_{rm WDM}$=0.5, 1 keV/$c^2$) relative to the CDM run. These differences are consistent with the mass assembly histories of the different components, and are present across redshifts. However, even the least discrepant of the WDM models is incompatible with current observational limits on $m_{rm WDM}$. Importantly, the differences we observe when varying the underlying dark matter are comparable to the galaxy-to-galaxy variation we expect within a fixed dark matter model. This suggests that it will be challenging to place limits on dark matter using only the unresolved spatial structure of the the ESC.
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