I never learned Elixir, but my first guess is IO.inspect is sending a message that is printed by a different process. Then the prints after exit are just the IO process working through its mailbox.
Alternatively, the await might killing the waiting process, not the process being waited on.
Good thoughts, but the printing continues indefinitely, and the documentation for Task.await explicitly says the child process will be killed: "If the timeout is exceeded, then the caller process will exit. If the task process is linked to the caller process which is the case when a task is started with async, then the task process will also exit". Processes can be configured with the behavior you describe, but it's not the case with Task.await.
Alternatively, the await might killing the waiting process, not the process being waited on.