[zeroer]

You may be right. I don't know the law that well.

Should there be a punishment for overstaying a visa? What should it be?

I think the answer may depend on the type of visa, the length of overstay, and perhaps some other factors, but in the majority of cases, I would expect the answer to be deportation, which would necessitate an arrest and holding. Whether that's called a criminal offense or civil offense is a distinction without a difference.