Yes, but for air that would require something in the order of 10000 atmospheres, or roughly 100km of water depth. So not on this planet. You would also need to add a lot of additional air until the water is saturated and stops dissolving it.
If you allow other liquids and gases, it becomes quite feasible. See this video of Cody: https://www.youtube.com/watch?v=AsP4yMY-a6U
Furthermore, an air bubble can be held in place under water e.g. by sound pressure. One particularly interesting instance is “single bubble sonoluminescence”, which (extremely simplified) uses sound to create a light-emitting bubble of steam and air. The first step is to [trap an air bubble in a spherical flask by sound](https://www.science.org/cms/10.1126/science.279.5355.1322/asset/e6ce1fe2-c05b-4f04-9a15-c21d42565635/assets/graphic/1322-1.gif). Either that or the vapour bubble then looks like [this](https://upload.wikimedia.org/wikipedia/commons/thumb/9/91/Single_bubble_cropped.jpg/800px-Single_bubble_cropped.jpg).
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