  Dark sector theories naturally lead to multi-component scenarios for dark
matter where a sub-component can dissipate energy through self-interactions,
allowing it to efficiently cool inside galaxies. We present the first
cosmological hydrodynamical simulations of Milky Way analogues where the
majority of dark matter is collisionless Cold Dark Matter (CDM), but a
sub-component (6%) is strongly dissipative minimal Atomic Dark Matter (ADM).
The simulations, implemented in GIZMO and utilizing FIRE-2 galaxy formation
physics to model the standard baryonic sector, demonstrate that the addition of
even a small fraction of dissipative dark matter can significantly impact
galactic evolution despite being consistent with current cosmological
constraints. We show that ADM gas with roughly Standard-Model-like masses and
couplings can cool to form a rotating "dark disk" with angular momentum closely
aligned with the visible stellar disk. The morphology of the disk depends
sensitively on the parameters of the ADM model, which affect the cooling rates
in the dark sector. The majority of the ADM gas gravitationally collapses into
dark "clumps" (regions of black hole or mirror star formation), which form a
prominent bulge and a rotating thick disk in the central galaxy. These clumps
form early and quickly sink to the inner ~kpc of the galaxy, affecting the
galaxy's star-formation history and present-day baryonic and CDM distributions.
