Fueled by inaccurate comments from Mr. Trump and others, election lies have spread across social media. They include claims that Black Lives Matter protesters incited violence at polling places, that mail-in ballots were dumped, that ballot boxes and voting machines were compromised, and that ballots were “harvested,” or collected and dropped off in bulk by unauthorized people.

Election officials in places such as Philadelphia, El Paso and Santa Rosa, Calif., are bearing the brunt of the fallout, according to interviews with a dozen of them in seven states. Some have had to contain misinformation-induced voter panic. Others are fighting back by posting accurate information on social media or giving newspaper and television interviews to spread their messages. Many are working longer shifts to debunk the distortions.

But their efforts have largely been fruitless, they said. When one rumor is smacked down, another pops up. And the reach of the rumors online is often so vast that the officials said they could not hope to compete.

“They’re definitely overwhelmed,” said Isabella Garcia-Camargo, an organizer of the Election Integrity Partnership, a new coalition of misinformation researchers.