I wouldn't really call that an atmosphere, so let's rephrase it: Can you have a ring of gas around a planet?
Yes. But not a very dense one. Jupiter, for example, has a plasma torus around it corresponding to the obit of Io, consisting of material from Io's atmosphere. It's not stable long-term, and has to be continually replenished by the loss of material from Io's surface... but that's true of Saturn's solid rings as well. They are a temporary structure which will last for a few hundred million years at most. And in fact, Saturn kinda already has a gas ring of its own, consisting of material ejected from Enceladus.
In all those cases, however, the gas tori are so diffuse that they would still be considered hard lab-grade vacuum here on Earth. They are so diffuse that the very existence of other examples of them in our solar system is debatable--there's some evidence that they exist, but they aren't the kind of dense structures that you can obviously see through a telescope!
So, can you have a dense ring of gas with significant internal pressure, which might reasonably be termed an "atmosphere", orbiting a planet? Well, kind of, although it is debatable whether the central object should really be called a "planet" in such cases, as they will necessarily be extremely massive. And the "gas ring" is likely to be more of a "liquid and gas ring", as it will also need to be massive enough to hold high-pressure gasses in place with its own self-gravity; i.e., you'll be looking at something like a toroidal gas-giant planet.
See, for example, General Relativistics Structure of Star-Toroid Systems. The math does not depend on the central object actually being a luminous star, but it does depend on the central object being massive--and the gas ring also being extremely massive. It cannot be stabilized only by the gravity of the central mass. Such structures have been observed in nature (see, for example, Hoag's Object), but only on galactic scales--not even the supposedly-stable stellar ones--so the possibility of something like that naturally forming on the scale of a planet seems Extremely Low.
If there was gas orbiting the planet and that gas reached down to the solid surface, if any, of the planet, then the solid surface of the planet would be scoured by winds travelling at orbital velocities, tens of thousands of miles per hour. Any life forms on the surface would be destroyed by the winds.
The winds would destroy every surface feature and would be slowed down to lower speeds by the collisions. So the lower air molecules would drop out of orbit and crash into the planet's surface at speeds slightly less than orbital speeds. Kaboom! So the only air molecules left in the orbital atmosphere would be orbiting the planet somewhat higher than the surface of the planet.
So it would be impossible for the orbital atmosphere to reach down to the surface. If the planet is to be habitable for humans, or have multi celled lifeforms like plants and animals, or intelligent natives, it has to have an atmosphere.
Thus the planet would need to have two different atmospheres. One reaching up from the surface of the planet and becoming thinner and thinner with altitude, and moving at the same speed as the planet rotates, and another atmosphere far above which travels at orbital speeds, probably in a torus above the planet's equator.
The two atmospheres can't touch, because the orbital speed of the orbital atmosphere should be many times as fast as the rotational speed of the planet. So if an orbiting air molecule hits a slower moving molecule from the lower atmosphere one or both molecules are likely to be knocked out of the zone where the two atmospheres meet, which should thus become cleared of air molecules.
as I remember, Larry Niven, who writes somewhat "harder" than usual science fiction, did write a novel, The Integral Trees, set in a gas torus of breathable air around a star. Thus it is possible that he carefully calculated that there could be a gas torus of breathable air in space that wouldn't swiftly dissipate. You should look for discussions of the scientific plausible of The Integral Trees to see how plausible a torus of dense gas in space is.
I suppose that it is theoretically possible for the lower atmosphere and the orbital atmosphere to meet under extreme conditions. Planets are mostly held together by their gravity. If a planet rotates too fast it will become more and more oblate, getting wider and flatter until it resembles a pancake, and if it rotates too fast it will fly apart when gravity is no longer strong enough to hold it together.
So possibly a planet that rotates fast enough and becomes pancake shaped as a result might rotate at orbital speed at its equator. Thus a hypothetical orbital atmosphere might possibly extend down to the planet's equator and interact with the lower atmosphere around the planet without the air molecules in the two atmospheres having different enough speeds that differ.the zone of interaction between the two atmospheres would be cleared out.