“Terminate with extreme prejudice” is a CIA play-on-words from the corporate phrases “terminated without prejudice” and “terminated with prejudice”, referring to firings that do and don’t leave the former employee free to reapply to the company if positions open up later, respectively.
In original usage, then, it only refers to killing people who are currently working for you, as intelligence agencies sometimes do.
That’s probably adapted from legal jargon, where dismissing a case with prejudice is a final resolution (either “the underlying matter is resolved” or “you are doing this with such bad faith/incompetence that we revoke your claim on more of this or any court’s time”) but dismissal without prejudice means “this is no good as is, but you can refile if things change”
