Q: Can a tidally-locked world support an atmosphere? I heard it would
all freeze out on the dark side, right?

A: It's a common idea that tidally-locked worlds can't have much of an
atmosphere, because the dark side would get so cold that any gases would
condense there, freezing the atmosphere onto the unlit side of the
planet. But the atmosphere can transport heat, lots of it, as can ocean
circulation is it's present. So, under what conditions can you have an
atmosphere that doesn't "collapse" (freeze out)? A simple global
circulation model was applied to the problem and generated some
surprising results. First, at Earth-normal levels of sunlight, both a
carbon dioxide atmosphere like Venus or a nitrogen dominated atmosphere
like Earth's would NOT freeze out. For a Earth-like planet sporting one
atmosphere of carbon dioxide, day-side temperatures would reach 330 K
(or 135 F), while the dark-side would be just below 270 K (25 F, just
below freezing) - at a slightly higher pressure, the dark-side
temperatures would be above the freezing point of water and global
oceans would even be possible. So while the dark-side can get cold, the
atmosphere would not come close to freezing out.
     Even more interesting is the winds: one simple assumption is that
warm day-side air would rise and flow back to the dark-side, cooling and
sinking there before flowing back across the terminator onto the
day-side (ground level winds blow towards the center of the day-side).
In fact the model showed that in the entire upper atmosphere
superrotates, rotating around the planet even faster than the planet
rotates, similar to Venus. Meanwhile the lower atmosphere has a very
different circulation, with warm day-side air streaming at ground level
across the equatorial terminators at mean speeds of 20 mph, cooling and
returning to day-side across the poles (equatorial winds blow away from
the day-side, while winds over the poles blow towards the day-side). So
heat is rapidly distributed across the whole surface, warming the
dark-side significantly and preventing atmospheric collapse.
     The model was also used to study how the atmosphere was affected by
the planet size, starspots (where the insolation of a M-type star can
drop by a factor of two for several months), or different compositions.
It was remarkably robust, and collapse was prevented even by some
extremely thin atmospheres (like 10 mb of carbon dioxide, similar to
Mars, under Earth-normal insolation). Liquid water oceans under a
breathable (nitrogen/oxygen) atmosphere might even be possible: the
dark-side must be warm enough not to trap all the water as ice, which
might make the day-side so warm that UV-photolysis and hydrogen escape
would lead to water loss - but M-type stars are so cool they have very
little energy shortwards of 0.2 micrometers, thus UV-photolysis might
not be significant.
     In short, not only can such a tide-locked planet maintain an
atmosphere, but it might even be habitable over much of its surface,
with an active water cycle and maybe even a near-breathable surface.

Source: Joshi, Haberle, & Reynolds, "Simulations of the Atmospheres of
Synchronously Rotating Terrestrial Planets Orbiting M Dwarfs: Conditions
for Atmospheric Collapse and the Implications for Habitability", Icarus
V129, pp450-465, 1997