TMZ has learned former NFL running back Chris Perry was arrested in Brooklyn yesterday, after cops say he got in a fist fight with a family member.

Law enforcement sources tell TMZ Perry -- a 2003 Heisman Trophy finalist -- was arrested in his grandmother's home after he allegedly socked his cousin in the face after an argument.

The arrest comes almost 6 years -- to the day -- from when the Michigan standout was selected in the first round of the NFL draft by the Cincinnati Bengals ... who ultimately cut the running back from the squad in 2009.

UPDATE: TMZ spoke with Perry's reps, who told us, "This is obviously a very difficult time for Chris personally as he's been dealing with his mother's terminal illness."

The reps added, "Tensions are obviously very high, as it would be for any family coping with a loved one suffering from cancer. Chris' strong relationship with his mother, and her current care, is his main focus."