  Observations suggest that satellite quenching plays a major role in the
build-up of passive, low-mass galaxies at late cosmic times. Studies of
low-mass satellites, however, are limited by the ability to robustly
characterize the local environment and star-formation activity of faint
systems. In an effort to overcome the limitations of existing data sets, we
utilize deep photometry in Stripe 82 of the Sloan Digital Sky Survey, in
conjunction with a neural network classification scheme, to study the
suppression of star formation in low-mass satellite galaxies in the local
Universe. Using a statistically-driven approach, we are able to push beyond the
limits of existing spectroscopic data sets, measuring the satellite quenched
fraction down to satellite stellar masses of ${\sim}10^7~{\rm M}_{\odot}$ in
group environments (${M}_{\rm{halo}} = 10^{13-14}~h^{-1}~{\rm M}_{\odot}$). At
high satellite stellar masses ($\gtrsim 10^{10}~{\rm M}_{\odot}$), our analysis
successfully reproduces existing measurements of the quenched fraction based on
spectroscopic samples. Pushing to lower masses, we find that the fraction of
passive satellites increases, potentially signaling a change in the dominant
quenching mechanism at ${M}_{\star} \sim 10^{9}~{\rm M}_{\odot}$. Similar to
the results of previous studies of the Local Group, this increase in the
quenched fraction at low satellite masses may correspond to an increase in the
efficacy of ram-pressure stripping as a quenching mechanism in groups.
