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The predominantly ancient stellar populations observed in the lowest mass galaxies (i.e. ultra-faint dwarfs) suggest that their star formation was suppressed by reionization. Most of the well-studied ultra-faint dwarfs, however, are within the central half of the Milky Way dark matter halo, such that they are consistent with a population that was accreted at early times and thus potentially quenched via environmental processes. To study the potential role of environment in suppressing star formation on the smallest scales, we utilize the Exploring the Local Volume in Simulations suite of N -body simulations to constrain the distribution of infall times for low-mass subhaloes likely to host the ultra-faint population. For the ultra-faint satellites of the Milky Way with star formation histories inferred from Hubble Space Telescope imaging, we find that environment is highly unlikely to play a dominant role in quenching their star formation. Even when including the potential effects of pre-processing, there is a |${\lesssim } 0.1{{\ \rm per\ cent}}$| probability that environmental processes quenched all of the known ultra-faint dwarfs early enough to explain their observed star formation histories. Instead, we argue for a mass floor in the effectiveness of satellite quenching at roughly |${M}_{\star }\sim 10^{5}\, {\rm M}_{\odot }$|⁠, below which star formation in surviving galaxies is globally suppressed by reionization. We predict a large population of quenched ultra-faint dwarfs in the Local Field (1 < R / R vir < 2), with as many as ∼250 to be discovered by future wide-field imaging surveys. [ABSTRACT FROM AUTHOR]