When someone is very busy at work, you can describe them as "swamped".
A: Hey, do you want to go out for lunch?
B: I can't today. I'm swamped.
The word "swamp" means a big pool of mud. So when you're "swamped" with work, it's like being stuck in a pool of mud that you can't get out of.
If someone is busy but not extremely busy, just use the word "busy" instead.