  We present the first set of cosmological baryonic zoom-in simulations of
galaxies including dissipative self-interacting dark matter (dSIDM). These
simulations utilize the Feedback In Realistic Environments (FIRE-2) galaxy
formation physics, but allow the dark matter to have dissipative
self-interactions analogous to Standard Model forces, parameterized by the
self-interaction cross-section per unit mass, $(\sigma/m)$, and the
dimensionless degree of dissipation, $0<f_{\rm diss}<1$. We survey this
parameter space, including constant and velocity-dependent cross-sections, and
focus on structural and kinematic properties of dwarf galaxies with $M_{\rm
halo} \simeq 10^{10-11} {\rm M}_{\odot}$. Central density profiles of simulated
dwarfs become cuspy when $(\sigma/m)_{\rm eff} \gtrsim 0.1\,{\rm
cm^{2}\,g^{-1}}$ (and $f_{\rm diss}=0.5$ as fiducial). The power-law slopes
asymptote to $\alpha \approx -1.5$ in low-mass dwarfs independent of
cross-section, which arises from a dark matter "cooling flow". Through
comparisons with dark matter only simulations, we find the profile in this
regime is insensitive to the inclusion of baryons. However, when
$(\sigma/m)_{\rm eff} \ll 0.1\,{\rm cm^{2}\,g^{-1}}$, baryonic effects can
produce cored density profiles comparable to non-dissipative cold dark matter
(CDM) runs but at smaller radii. Simulated galaxies with $(\sigma/m) \gtrsim
10\,{\rm cm^{2}\,g^{-1}}$ develop significant coherent rotation of dark matter,
accompanied by halo deformation, but this is unlike the well-defined thin "dark
disks" often attributed to baryon-like dSIDM. The density profiles in this high
cross-section model exhibit lower normalizations given the onset of halo
deformation. For our surveyed dSIDM parameters, halo masses and galaxy stellar
masses do not show appreciable difference from CDM, but dark matter kinematics
and halo concentrations/shapes can differ.
