“time limit,” 1920, American English newspaper jargon, from dead (adj.) + line (n.). Perhaps influenced by earlier use (1864) to mean the “do-not-cross” line in Civil War prisons, which figured in the trial of Henry Wirz, commander of the notorious Confederate prison at Andersonville, Georgia.
And he, the said Wirz, still wickedly pursuing his evil purpose, did establish and cause to be designated within the prison enclosure containing said prisoners a “dead line,” being a line around the inner face of the stockade or wall enclosing said prison and about twenty feet distant from and within said stockade; and so established said dead line, which was in many places an imaginary line, in many other places marked by insecure and shifting strips of [boards nailed] upon the tops of small and insecure stakes or posts, he, the said Wirz, instructed the prison guard stationed around the top of said stockade to fire upon and kill any of the prisoners aforesaid who might touch, fall upon, pass over or under [or] across the said “dead line” … [“Trial of Henry Wirz,” Report of the Secretary of War, Oct. 31, 1865]
Check out deadline.