Wiktionary
vb. (context euphemistic US English) To murder; to assassinate.
Wikipedia
In military and other covert operations, terminate with extreme prejudice is a euphemism for aggressive execution (playing on the expression " termination with prejudice" of an employment contract). In a military intelligence context, it is generally understood as an order to assassinate. Its meaning was explained in a New York Times report on the Green Beret Affair (an incident during the Vietnam War), and subsequently popularized by its usage in the film Apocalypse Now.
The expression may also be used as a straightforward statement of intent in some jurisdictions to refer to more overt summary executions, i.e. in the context of a 'death warrant' or 'kill order' issued as part of a 'shoot on sight' or similar policy.
Usage examples of "terminate with extreme prejudice".
After hitting Cheesewright he explained carefully that Terminate with Extreme Prejudice did not simply require that the victim was inhumed, preferably in an extremely thorough way, but that his associates and employees were also intimately involved, along with the business premises, the building, and a large part of the surrounding neighbourhood, so that everyone involved would know that the man had been unwise enough to make the kind of enemies who could get very angry and indiscriminate.
It was no time for mercy, it was time to terminate with extreme prejudice.