Smart will seek a second opinion this week to determine the severity of the tendon damage and whether surgery is needed.
Smart scored 20 points with eight assists and seven rebounds Sunday in a loss to the Pacers. Surgery would probably mean a lengthy absence for Smart, who is a key part to Boston's success as their first guard off the bench.
The Celtics will play the Wizards on Wednesday.
But now he's one of the many players on the mend, joining Kyrie Irving (knee), Jaylen Brown (concussion), Al Horford (illness), and Shane Larking (knee) on Boston's injured list. Smart missed time earlier in the season after lacerating that same hand when he hit a picture frame on the wall at the team hotel during an angry outburst. Though he initially said he would be "all right", it's clear the Celtics now fear he could miss extended time.