I'm referring to simply being in the country without authorization, or having entered illegally; what is called in the US "undocumented", and which most countries treat with deportation.
ChatGPT tells me that Saudi Arabia and Iran execute people for "immigration-related offences" such as drug trafficking, so that doesn't count yet.
Are there any countries that impose the death penalty for being in the country illegally?
No.
Sometimes illegal immigrants are denounced as "spies" which can be a death penalty offense in places like North Korea. But illegal immigration per se is not a death penalty offense in any country having immigration-type laws. (There are some basically un-contacted tribes in the Andaman Islands or the Amazon or Papua New Guinea that kill strangers on sight without any legal process.)
