  We present the first cosmological simulations of dwarf galaxies, which
include dark matter self-interactions and baryons. We study two dwarf galaxies
within cold dark matter, and four different elastic self-interacting scenarios
with constant and velocity-dependent cross sections, motivated by a new force
in the hidden dark matter sector. Our highest resolution simulation has a
baryonic mass resolution of $1.8\times 10^2\,{\rm M}_\odot$ and a gravitational
softening length of $34\,{\rm pc}$ at $z=0$. In this first study we focus on
the regime of mostly isolated dwarf galaxies with halo masses
$\sim10^{10}\,{\rm M}_\odot$ where dark matter dynamically dominates even at
sub-kpc scales. We find that while the global properties of galaxies of this
scale are minimally affected by allowed self-interactions, their internal
structures change significantly if the cross section is large enough within the
inner sub-kpc region. In these dark-matter-dominated systems, self-scattering
ties the shape of the stellar distribution to that of the dark matter
distribution. In particular, we find that the stellar core radius is closely
related to the dark matter core radius generated by self-interactions. Dark
matter collisions lead to dwarf galaxies with larger stellar cores and smaller
stellar central densities compared to the cold dark matter case. The central
metallicity within $1\,{\rm kpc}$ is also larger by up to $\sim 15\%$ in the
former case. We conclude that the mass distribution, and characteristics of the
central stars in dwarf galaxies can potentially be used to probe the
self-interacting nature of dark matter.
