He was almost immediately ruled out for the remainder of the game with a left hand injury, and the team announced the specifics of his injury before halftime.
Baynes did not speak to reporters after the 111-103 loss to the Suns.
"I don't know any updates on timelines or anything like that," Celtics coach Brad Stevens said. "But, probably, clearly, he's going to be out for some time."
Stevens said Baynes was "disappointed."
The loss of Baynes further reduces Boston's depth in the middle, where the Celtics already have been dealing with Horford missing the past five games with patellar tendinitis in his left knee.
Before Wednesday's game against the Suns, Stevens said Horford had made some progress and had done some on-court work but that the team was going to remain cautious with him in his recovery.
"I think anytime -- even looking behind at the schedule we just played and now having a few days off, as you're preparing to get on the practice court and walkthroughs and those type of things -- you're thinking about what you just finished and what's coming," Stevens said. "This is an intense schedule all the way through All-Star break starting tonight, and I think that, ultimately, we want as many guys feeling good as possible.
"Everybody's got nicks and bruises, but if they're not feeling really, really good, then we'll hold them out."
Baynes, 32, is averaging 5.5 points and 4.5 rebounds in just under 15 minutes per game for Boston.