"I'm just me" is totally incorrect. An error can be charged for prolonging a hitter's at bat, such as dropping a foul fly ball or popup...
An error is a statistic charged against a fielder whose action has assisted the team on offense, as set forth in this Rule 10.12.
(a) The official scorer shall charge an error against any fielder:
(1) whose misplay (fumble, muff or wild throw) prolongs the time at bat of a batter, prolongs the presence on the bases of a runner or permits a runner to advance one or more bases, unless, in the judgment of the official scorer, such fielder deliberately permits a foul fly to fall safe with a runner on third base before two are out in order that the runner on third shall not score after the catch;
(the above paragraph is followed in the official rules by a long comment, not reproduced in this answer)
(2) when such fielder muffs a foul fly to prolong the time at bat of a batter, whether the batter subsequently reaches first base or is put out;
________________________________________
Perfect games and No-hitters:
An official perfect game occurs when a pitcher (or pitchers) retires each batter on the opposing team during the entire course of a game, which consists of at least nine innings. In a perfect game, no batter reaches any base during the course of the game.
___________________________________________
So in the case of a fielder dropping a foul ball which prolongs an at bat, he can be charged with an error. But if the hitter subsequently makes an out, the perfect game would still be intact.
A perfect game is simply defined as a game in which a pitcher does not allow an opposing batter to reach base in any manner.
Therefore, a perfect game most certainly can have an error occur (at least theoretically) as long as no opposing batter reaches base.