MAIDAN SHAHR, Afghanistan — An Afghan policeman gunned down two U.S. special forces Monday in Wardak province, less than 24 hours after President Hamid Karzai’s deadline expired for them to leave the area where residents have grown increasingly hostile toward the Americans.

Despite Karzai’s orders, the American special operations forces remain in the province, where dozens of villagers accuse them and their Afghan partners of intimidation through unprovoked beatings, mass arrests and forced detentions. The shootout, which also killed two Afghan policemen, only deepens the distrust.

The U.S.-led coalition in Afghanistan says it has found no evidence to support the claims of abuse. But infuriated by the villagers’ allegations, Karzai two weeks ago ordered U.S. special operations forces to withdraw by midnight Sunday from Wardak province, 27 miles south of the capital, Kabul.

Most international forces are scheduled to withdraw from Afghanistan by the end of 2014. Wardak, like the rest of the country, is due to be eventually handed over to Afghan forces, but U.S. Gen. Joseph Dunford, the top commander of U.S. and coalition forces in Afghanistan, indicated Sunday that the troops were not leaving Wardak province just yet.

Wardak has a stubborn insurgency on the doorstep of the capital Kabul and its location has led some U.S. military officials to warn that a premature withdrawal of U.S. special operations forces would open a “six-lane highway” into Kabul for the Taliban.

But Afghan security forces disagree, saying they don’t think insurgents can capture the provincial capital.

On Monday, an Afghan policeman stood up in the back of a pickup truck, grabbed a machine gun and started firing at U.S. special forces and other Afghan policemen at a police compound in Wardak’s Jalrez district, about 12 miles east of Maidan Shahr, said the province’s Deputy Police Chief Abdul Razaq Koraishi.

Two U.S. special operations forces and two Afghan policemen were killed and four others were wounded in the gunfight before the assailant was gunned down, Koraishi said.

A U.S. defense official in Washington and a coalition official in Afghanistan said 10 Americans and at least 12 Afghans were wounded in the attack.