“This is about religious freedom,” she said. “This is not about trashing gay people.”

But Ms. DiBiaggio and Ms. Carmichael say it is discriminatory, and they are contemplating filing a lawsuit. The women, who have been together for 10 years and own a working farm in nearby Springport, declined Ms. Belforti’s request that they make an appointment and return later when they stopped by the Ledyard clerk’s office seeking a marriage license.

“I was shocked,” Ms. Carmichael said.

Deborah Liu, the general counsel for the People for the American Way Foundation, which is working with a New York law firm, Proskauer Rose, on the case, said, “We totally respect everyone’s right to have their own personal beliefs.” But Ms. Belforti, Ms. Liu said, “doesn’t have the right to use them to relieve herself from doing a major part of her duties.”

The issue has roiled this town, where Republicans slightly outnumber Democrats, and often on Election Day there are fewer voters than for a race for high school class president. Most of the town is farmland, but Ledyard also contains the village of Aurora, home to Wells College, an opera house and a lakeside inn where a one-night wine country getaway goes for as much as $813 per couple.

Image Ledyard is a farming town with 1,900 residents. Credit... The New York Times

In interviews last week, some residents applauded Ms. Belforti for standing up for her beliefs, while others said she should either sign all marriage licenses or find another line of work.

“You get about a 50-50 split,” said Jim Wilcox, 42, who runs the white-clapboard Wilcox General Store and who got his own marriage license from Ms. Belforti a few years ago. He called the Marriage Equality Act “a long time coming” and worried that the controversy could paint Ledyard in a bad light. “We’d hate to be the ones who slowed down the wheels of change here,” he said.

Another resident, Ed Easter, is now seeking to defeat Ms. Belforti in a write-in campaign when she is up for re-election in November. Mr. Easter, 40, who works in a wine tasting room, said he felt that someone needed to challenge her, rather than assuming the courts would eventually settle the matter.