To surround someone or something with something overpowering or pervasive, especially an odor. Often used in passive constructions. His cooking swamped the house with the smell of onions and garlic. I walked in the door and was instantly swamped with the stale stench of cigarettes, booze, and unwashed men.

Are you swamped with work?

The phrase swamped with work is an informal way to say that one has an excess of things to do.

What does get swamped mean?

If you're swamped, it means you're very busy and have a lot of work.