We present FIRE/Gizmo hydrodynamic zoom-in simulations of isolated dark matter halos, two each at the mass of classical dwarf galaxies ($M_{rm vir} simeq 10^{10} M_{odot}$) and ultra-faint galaxies ($M_{rm vir} simeq 10^9 M_{odot}$), and with two feedback implementations. The resultant central galaxies lie on an extrapolated abundance matching relation from $M_{star} simeq 10^6$ to $10^4 M_{odot}$ without a break. Every host is filled with subhalos, many of which form stars. Our dwarfs with $M_{star} simeq 10^6 M_{odot}$ each have 1-2 well-resolved satellites with $M_{star} = 3-200 times 10^3 M_{odot}$. Even our isolated ultra-faint galaxies have star-forming subhalos. If this is representative, dwarf galaxies throughout the universe should commonly host tiny satellite galaxies of their own. We combine our results with the ELVIS simulations to show that targeting $sim 50~ rm kpc$ regions around nearby isolated dwarfs could increase the chances of discovering ultra-faint galaxies by $sim 35%$ compared to random halo pointings, and specifically identify the region around the Phoenix dwarf galaxy as a good potential target. The well-resolved ultra-faint galaxies in our simulations ($M_{star} simeq 3 - 30 times 10^3 M_{odot}$) form within $M_{rm peak} simeq 0.5 - 3 times 10^9 M_{odot}$ halos. Each has a uniformly ancient stellar population ($ > 10~ rm Gyr$) owing to reionization-related quenching. More massive systems, in contrast, all have late-time star formation. Our results suggest that $M_{rm halo} simeq 5 times 10^9 M_{odot}$ is a probable dividing line between halos hosting reionization fossils and those hosting dwarfs that can continue to form stars in isolation after reionization.
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