Atmolayer Lip Hypothesis
The Flat Earth does not necessarily need to be physically infinite in order to contain the atmosphere. Just very big. Often we might hear "infinite earth" from Flat Earth proponents as an analogy for what exists beyond the 150 foot wall of ice at the Antarctic coast; a stretch of land incomprehensible by human standards.
In order for barometric pressure to rise and fall, an element of heat must be present. Heat creates pressure. A lack of heat results in a drop in pressure. These two elements are tightly correlated in modern physics.
In our local area the heat of the day comes from the sun, moving and swashing around wind currents between areas of low pressures and areas of high pressures with its heat. Beyond the known world, where the rays of the sun do not reach, the tundra of ice and snow lays in perpetual darkness. If one could move away from the Antarctic rim into the uncharted tundra the surrounding temperatures, and therefore pressures, would drop lower and lower. Defining the exact length of the gradient would take some looking into, but at a significant distance past the edge of the Ice Wall temperatures will drop to a point where barometric pressure nears the zero mark. At this point, whether it be thousands or millions of miles beyond the Antarctic rim, the environment will gradually match that of background space, and the world can physically end without the atmosphere leaking out of it.
The atmosphere may very well exist as a lip upon the surface of the earth, held in by vast gradients of declining pressure.
Freezing Balloons Video
See this interesting video about what happens to balloons when they are frozen:
Link: https://www.youtube.com/watch?v=9vRMZSEF_a4
Questions and Answers
Q. From some sources is appears that Antarctica at sea level is quite a high pressure area, which doesn't play well with the idea of beyond it being increasingly lower pressure areas.
A. It has been demonstrated through a number of experiments that in a closed system there is a correlation between heat and barometric pressure. However, the atmosphere is not a container. Air is free to rise, cool, fall, ect. The behavior of the climate is very complex, but over a greater distance, beyond the heating influences of the sun's light, the behavior would presumably more closely match what is seen in a closed system.