PI can also easily generate perfect pitching performances (allowing no baserunners). Requiring 5 IP or more, this gives the usual perfect complete games list (since 1956), but also includes other good outings, often by relievers.
Mark Williamson's game on 1989-04-12 is interesting in that he got a BS for pitching 5 perfect innings, the only such game on the list with a bad outcome for the pitcher (L or BS). He came on in the 9th with the bases loaded and no outs and induced a GDP, but the tying run scored. I think this is also the longest perfect pitching performance since 1956 where the BF is less than 3 times the IP (thanks to double plays, etc.). That's harder to check, but I checked individually all possible numbers of IP (including thirds of innings) up to 13, so I think it's right.

The only pitcher since 1956 to allow no base runners and lose the game is
Bill Landis on 1969-06-29. Think about it, he didn't allow a runner and yet was credited with the loss...! What happened was he pitched to one batter, who got on by FC. On the play Landis himself made throwing error attempting to make a force out at 2nd or 3rd base. Landis was then removed from the game, but the player who got on by FC was eventually the winning run for the other team, giving Landis the loss. I guess a runner who gets on by a FC is not counted as a BR for the pitcher? That makes sense, since usually on a FC someone else is put out, and the number of men on base does not increase. The extra man on base was due to the error. And I suppose an error by anyone would have had the same result, but it's poetic justice that it was Landis' error that lead to his getting the loss.

Ron Hassey only caught 2 no-hitters, but they were both perfect games.

The '03 Yankees also won the pennant. As did the '90 A's, and the '81 Dodgers, both of which were no-hit. I stopped checking after that. I doubt it's so unlikely for a no-hit team to win the pennant, any more unlikely than that they get shutout -- it's only one game after all. Wouldn't expect it to happen often but obviously it's not unique.

An interesting thing about the 1958 no-hitter against the Yankees is that the pitcher was Hoyt Wilhelm, in one of the few years that he was used much as a starter. Another interesting thing is that the Yankees won the pennant despite being no-hit - that doesn't happen a lot.

I have a book that lists the opposing pitcher for each (MLB definition) no-hitter up to 1992. It doesn't say whether "opposing pitcher" means "opposing starter" or "guy who got the decision for the other team." I can confirm that Jim Perry was the opposing pitcher in three no-hitters, as noted above. I found several men who were opposing pitchers in two no-hitters: Ed Walsh, Slim Harriss, Chuck Dobson, Charlie Hough, Tommy Bond, Chris Short, Phil Niekro, and Vida Blue. Blue is the only one I found who was an opposing pitcher in a no-hitter in each league: Jim Bibby, Tex vs Oak, 30 July 73, and Jerry Reuss, LA vs SF, 27 Jun 80.

Eight teams have been no-hit twice in one season: 1960 Phillies, 1965 Cubs, 1967 Tigers, 1971 Reds, 1973 Tigers, 1977 Angels, 1996 Rockies, and 2001 Padres. No team has been no-hit three times in one season.

The Yankees have had the longest team hitting streak (45 years), having gone from 1958 to 2003 without being on the short end of a no-no.

The longest current streak is held by the Cubs, who haven't been no-hit since Koufax's perfect game in 1965 (43 years and counting). The Reds and Pirates are at 37 years, and the Phillies are at 30 years.

The Cubs owe a lot to Billy Williams for keeping that streak alive. Five times in his Cubs career, Williams was the only player on his team to get a hit in a game. And in one of those games (09-05-69 vs. Pittsburgh), Williams got the only FOUR hits off of Steve Blass! (That has to be a record, yes? Most hits by one player when the rest of his team took a collective oh-fer?)

Varitek now has 4 official no-hitters (Nomo, Lowe, Buchholz & Lester). He's got a 5th that's unofficial, as Devern Hansack's 5 inning no-hitter against the Orioles doesn't count.