Such an play occurred on August 2, 1985. The score was 3-3 in the bottom of the seventh when Andre Robertson led off the inning with a single to center. Bobby Meacham went in to run for Robertson. Dale Berra was then safe on Tim Hulett’s error, Meacham stopping at second. Rickey Henderson followed with a single to left-center. Luis Salazar threw the ball to Ozzie Guillen who relayed the ball to Carlton Fisk to catch both Meachum and Berra. New York would end up losing the game 6-5 in 11 innings. (It wasn’t an unassisted DP because both Salazar and Guillen had assists on the play.)
There was a similar play involving Buck Martinez and the Seattle Mariners, which occurred in the bottom of the third inning of the July 9, 1985 game between the Blue Jays and the Mariners. Phil Bradley was on second with one out when Gorman Thomas hit a single to right. Jesse Barfield threw to Buck Martinez to nail Bradley. Martinez broke his ankle in the collision. When Thomas attempted to take third on the play, Martinez’ throw sailed into left field. Thomas headed home but George Bell’s throw beat him, with Martinez making the catch and tag while sitting on the ground. Not quite the bang-bang play involving Fisk, but weird enough nonetheless.
Since 1980, a catcher has made two putouts (not including strikeouts) on the same play one other time. That occurred in Montreal on April 18, 1992. The visiting Mets had Dave Gallagher on second and Willie Randolph on first with one out in the top of the sixth inning when Mackey Sasser hit a single to right. Larry Walker threw home to Darrin Fletcher to nail Gallagher (who forgot to slide). Fletcher threw to first, trying to catch Sasser wandering off the bag, but the ball hit Mackey instead. Randolph then tried to score but was thrown out, the play eventually being scored 9-2-6-3-2.