davidmarver is very close...but he fell for my trap!
The Pirates played a 21 inning game where one pitcher (Babe Adams) walked no one. I didn't say it had to be one pitcher throwing, though.
The Montreal Expos played a 22 inning game in 1989 against the Dodgers where none of the pitchers in the game allowed a walk. They lost the game 1-0 on a Rick Dempsey walkoff HR. He was my original query - I loved his rainout act!
Here's the writeup:
August 23, 1989: Rick Dempsey homers off Dennis Martinez in the top of the 22nd inning to break up a scoreless tie and give the Dodgers a 1–0 win over the Expos. The game features one thumbing -- the umps toss the Expos mascot Youppi in the 11th for annoying Tommy Lasorda -- and he then returns in the 13th wearing pajamas. He carries a pillow and sleeps on the home dugout roof, where the umps have restricted him. In the 16th, Larry Walker apparently scores the game-winner, but the Dodgers appeal -- with two umps in the tunnel -- and get it. Eddie Murray in the 18th moves the 2B ump and slams a drive that Walker makes a phantom catch off the padding in RF. Fans stand for three "seventh-inning stretches" during the major-league record (in time, and for a 1–0 game) six hours: 14 minute game. Other club records are set and several ML records are noted: most innings (22) without a walk by the Expos tops the Pirates (who used one pitcher) against the Giants, July 17, 1914.
The One, The Only, The Incomparable Mercer Boy.
blog (updated 6/21 - Pirates walkout.)