A perfect game that is not a no-hitter?

[Hallas]

Hypothetical question: Say a batter hits a single but is thrown out trying to stretch to a double. The batter never reaches base, having been thrown out in the same play that he hit the ball in play. If this is the only hit in a game, does the game still count as a perfect game?

The MLB rules specifically say 27 up, 27 down, with no distinction as to how the 27 batters are thrown out. However, other sources seem to indicate that a perfect game is by necessity a no-hitter.

This also applies to ROEs where the batter is thrown out trying for an extra base.