Here it is black humour. To “meet a deadline” has its roots in 19th century warfare and is surprisingly literal in its origins.
Most etymologists agree that the word “deadline” first appeared during the American Civil War (1861-1865). According to Christine Ammer, deadline was coined at the hellish Andersonville, GA prison camp, and first appeared in writing in the report of Confederate Inspector-General, Colonel D.T. Chandler, on July 5, 1864. In describing the horrific conditions, he famously wrote:
"The Federal prisoners of war are confined within a stockade 15 feet high, of roughly hewn pine logs, about 8 inches in diameter, inserted 5 feet into the ground, enclosing, including the recent extension, an area of 540 by 260 yards. A railing around the inside of the stockade, and about 20 feet from it, constitutes the “deadline,” beyond which the prisoners are not allowed to pass . . . [as a large portion is] at present unfit for occupation . . . [this] gives somewhat less than 6 square feet to each prisoner ...."
Having said that, the context of a due date probably originated in journalism, perhaps from an earlier usage in printing, representing a guideline marked on a plate for a printing press (inside which all content should appear). According to the OED, early usage refers simply to lines that do not move, such as one used in angling; the American usage indicated probably came later.
