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21 Savage is a prisoner of America's immigration system -- he's facing deportation, he can't leave the country, his ability to make a living is severely impaired and he doesn't even have a court date.

As TMZ reported ... Savage was arrested in Atlanta back in February and placed in ICE custody. Turns out he entered the U.S. legally from the UK back in 2005, but his visa expired and he's been here illegally for years.

The rapper wasn't even able to get a work permit for 8 months, which meant he couldn't work domestically. He finally got one last month, so at least now he can work in the U.S. ... but traveling out of the country is out of the question until he gets things cleared up with immigration.

As you know, selling records doesn't cut it anymore, so artists make their money by doing concerts, and every country out of the U.S. is off-limits to him.

Here's the underlying problem ... we're told immigration courts in the ATL are ridiculously overcrowded with hundreds of backlogged cases. There are people who have gotten court dates for 2022. Savage hasn't even gotten a court date, so it could take years to resolve his case.