Marines at Normandy -- not exactly
When the Rangers were trapped at Pointe du Hoc (they accomplished their mission, but were cut off from the main US forces at Omaha Beach for 2-3 days) one Marine officer afloat wanted to reinforce them with the shipboard detachment of USS Texas.
Omar Bradley, personally, shot it down; he didn't want to see any headlines reading "US Army rescued by Marines!"
No Marines were put ashore on D-Day, and I think few thereafter.
The tale about the Marine behind enemy lines sounds like Peter Ortiz. "Wild Bill" Donovan's OSS was a law unto itself (seeing as how Donovan was an old classmate and friend of some guy working out of a wheelchair in an Oval Office) and had free rein to pick anyone it chose. It got a few Marines. Peter Ortiz was, apparently, a good operative and USMC Captain; he did drop behind enemy lines and was captured by the Germans, surviving months not in a PoW camp but concentration camp. [Hitler had decreed that all "commandos" and similar special operatives be executed on capture; the German officers in the field decided that sending them to a concentration & labor camp was the most efficient way to do so. A fair number of British captured operatives survived such ordeals.]