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We analyze the first set of cosmological baryonic zoom-in simulations of galaxies in dissipative self-interacting dark matter (dSIDM). The simulations utilize the FIRE-2 galaxy formation physics with the inclusion of dissipative dark matter self-interactions modelled as a constant fractional energy dissipation ($f_{\rm diss}=0.5$). In this paper, we examine the properties of dwarf galaxies with $M_{\ast} \sim 10^{5}\operatorname{-}10^{9}\,{\rm M}_{\odot}$ in both isolation and within Milky Way-mass hosts. For isolated dwarfs, we find more compact galaxy sizes and promotion of stellar/neutral gas disk formation in dSIDM with $(σ/m)\leq 1\,{\rm cm^2\,g^{-1}}$ but they are still consistent with observed galaxy sizes and masses. In addition, as a result of the steeper central density profiles developed in dSIDM, the sub-kpc circular velocities of isolated dwarfs in models with $(σ/m)\geq 0.1\,{\rm cm^2\,g^{-1}}$ are enhanced by about a factor of two, which are still consistent with the measured stellar velocity dispersions of Local Group dwarfs but in tension with the HI rotation curves of more massive field dwarfs. Meanwhile, for satellites of the simulated Milky Way-mass hosts, the median circular velocity profiles are marginally affected by dSIDM physics, but dSIDM may help address the missing compact dwarf satellites in CDM. The number of satellites is slightly enhanced in dSIDM, but the differences are small compared with the large host-to-host variations. In conclusion, the dSIDM models with constant cross-section $(σ/m) \gtrsim 0.1\,{\rm cm^2\,g^{-1}}$ (assuming $f_{\rm diss}=0.5$) are effectively ruled out in bright dwarfs ($M_{\rm halo}\sim 10^{11}\,{\rm M}_{\odot}$) by circular velocity constraints. However, models with lower effective cross-sections (at this halo mass/velocity scale) are still viable and can give rise to non-trivial observable signatures.