ICE says 21 Savage's Public Persona Was Totally 'False'

21 Savage is a living, breathing example of that ol' saying, "fake it til you make it" ... at least according to ICE.

An ICE official told CNN on Sunday ... "His whole public persona is false. He actually came to the U.S. from the U.K. as a teen and overstayed his visa."

The arrest has totally shocked 21 Savage's fans ... who had NO CLUE about his British roots. We did some digging and to be fair, Sav has never claimed Atlanta as his birthplace. In multiple profiles on the rapper ... media outlets have referred to him as being "from" Atlanta or say he "grew up" there.

That said, it sure doesn't seem like he's ever gone out of his way to correct any reports of him being an ATL native. 21 definitely reps Atlanta in his rhymes ... and he's told tales about getting expelled from an Atlanta school for bringing a gun to class.

Still, he's never said he was "born" there.

As we reported ... Sav was arrested early Sunday in Atlanta by what ICE described as a targeted operation. ICE officials say 21 Savage is actually a U.K. national and has been illegally in the U.S. since 2006. They say he came here as a kid in 2005.