'Alcatraz' TV Show -- Delusional Fans Trying to Break In to Prison

Some fans of the TV show "Alcatraz" aren't so bright -- because they're infiltrating guided tours on the real-life island to search for a top secret room ... that only exists on the scripted drama!

National Parks Service rep Alexandra Picavet tells TMZ ... tourists are starting to stray from their tours and sneak into "closed areas" on the former federal prison.

Picavet says many of the not-so-super sleuths have confessed they're looking for a high tech control room (below) ... described on the show as a "bat-cave underneath Alcatraz."

Keep in mind the show just debuted last week. Crazy spreads quickly.

Picavet says they've now posted the following sign on the island: "The TV show Alcatraz is fictional, many areas it depicts are not real. Closed areas protect you, historic structures and nesting birds."

Parks Service peeps assure us the room doesn't exist. They also say the Easter Bunny and Tooth Fairy aren't real, so ... believe what you want.