The Golds, who are the last side from Sussex left in the competition, will visit Essex Senior League side Athletic Newham.
Fourth round ties will take place on Saturday, January 15. Winning clubs will receive £1,875 while losing teams will get £600.
The Golds are so far enjoying a stellar season. Littlehampton sit second in the SCFL Premier, two point behind leaders Saltdean United with two games in hand on their title rivals.