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TMZ Sports obtained police footage from the night NFL cornerback Damian Swann was arrested in Georgia in which he throws a serious Hail Mary ... asking the cop to give him a break because he plays for the New Orleans Saints!

The 23-year-old was pulled over in Athens-Clarke County, Georgia on March 18th -- with passenger Rico Johnson (an ex-Georgia football player) when cops said he was going 100 in a 65.

Swann told cops he had a firearm in his pocket and another gun in his car (Glock .380 and a S+W .40) -- but that didn't seem to be a problem as he had a valid Concealed Carry Permit.

But when the cop tells Swann he needs to take him into custody, the NFL player did everything he could to try to talk the officer out of it ... to no avail.