In general yes, but the code in this case appears to be supplying its own version of exit, that just does a SYSCALL instruction. The SYSCALL instruction doesn't need a stack.

Is there no risk of an interrupt or a signal happening in between un-mmap'ing and calling exit, that would cause stack pushes?

That is a risk, which is why you need to block all signals using the sigprocmask syscall before unmapping your stack.

Or you just don't have signal-handling functions. Signals only touch the stack if you handle them via functions. The default signal dispositions all either kill the process or do nothing, so they don't touch the stack.

It is calling the exit syscall directly using the syscall instruction, which does not push anything onto the stack.