We present deep Hubble Space Telescope imaging of five faint dwarf galaxies associated with the nearby spiral NGC 253 (D$approx$3.5 Mpc). Three of these are newly discovered ultra-faint dwarf galaxies, while all five were found in the Panoramic Imaging Survey of Centaurus and Sculptor (PISCeS), a Magellan$+$Megacam survey to identify faint dwarfs and other substructures in resolved stellar light around massive galaxies outside of the Local Group. Our HST data reach $gtrsim$3 magnitudes below the tip of the red giant branch for each dwarf, allowing us to derive their distances, structural parameters, and luminosities. All five systems contain predominantly old, metal-poor stellar populations (age$sim$12 Gyr, [M/H]$lesssim$$-$1.5) and have sizes ($r_{h}$$sim$110-3000 pc) and luminosities ($M_V$$sim$$-7$ to $-12$ mag) largely consistent with Local Group dwarfs. The three new NGC 253 satellites are among the faintest systems discovered beyond the Local Group. We also use archival HI data to place limits on the gas content of our discoveries. Deep imaging surveys such as our program around NGC 253 promise to elucidate the faint end of the satellite luminosity function and its scatter across a range of galaxy masses, morphologies, and environments in the decade to come.
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