Based on the literal meaning of snowed under ( covered by so much snow that people cannot get out or move around) Having too much work to do. Overworked; exceptionally busy. To overwhelm. To defeat by a very large margin. Examples; I was snowed under with work. She wants me to take some time off but I'm snowed under with work at the moment. We're completely SNOWED UNDER at work because it's the end of the tax year.

Be snowed under with something (especially work) is an idiom meaning 'to have too much work': 'I'm snowed under with work at the moment.' 'I'm afraid I can't come. I'm really snowed under at the moment.'

As though you were covered in snow... except the "snow" is "work".

It means to be so busy, it is impossible to do a task. Johns friend asked him to dinner, but John said he was snowed under with work, and he would have to have dinner on another night.