  A census of the satellite population around dwarf galaxy primary hosts in
environments outside the Local Group is essential to understanding $\Lambda$CDM
galaxy formation and evolution on the smallest scales. We present deep optical
Hubble Space Telescope imaging of the gas-rich, faint dwarf galaxy Antlia B
($M_V = -9.4$) -- a likely satellite of NGC 3109 ($D = 1.3$ Mpc) -- discovered
as part of our ongoing survey of primary host galaxies similar to the
Magellanic Clouds. We derive a new tip of the red giant branch (TRGB) distance
of $D = 1.35 \pm 0.06$ Mpc ($m-M = 25.65 \pm 0.10$), consistent with membership
in the nearby NGC 3109 dwarf association. The color-magnitude diagram shows
both a prominent old, metal-poor stellar component and confirms a small
population of young, blue stars with ages $\lesssim 1$ Gyr. We use the
color-magnitude diagram fitting algorithm MATCH to derive the star formation
history and find that it is consistent with the typical dwarf irregular or
transitional dwarf galaxy (dTrans) in the Local Group. Antlia B shows
relatively constant stellar mass growth for the first $\sim 10-11$ Gyr and
almost no growth in the last $\sim 2-3$ Gyr. Despite being gas-rich, Antlia B
shows no evidence of active star formation (i.e., no H$\alpha$ emission) and
should therefore be classified as a dTrans dwarf. Both Antlia B and the Antlia
dwarf (dTrans) are likely satellites of NGC 3109 suggesting that the cessation
of ongoing star formation in these galaxies may be environmentally driven.
Future work studying the gas kinematics and distribution in Antlia B will
explore this scenario in greater detail. Our work highlights the fact that
detailed studies of nearby dwarf galaxies in a variety of environments may
continue to shed light on the processes that drive the star formation history
and evolution of dwarf galaxies more generally.
