The attack "hit at the very basis of cooperation" between Pakistani and coalition forces in Afghanistan, the country's military said in a statement.

The prime minister, Yousaf Raza Gilani, condemned the attack. "We will take a stand for sovereignty, integrity and self-respect and we will not allow our soil [to be attacked]," he told Pakistan's parliament.

At least nine civilians were also killed, Afghan politicians said, during prolonged skirmishes and air strikes along Afghanistan's eastern border.

The air strikes followed a clash between Afghan forces and Taliban militants. There were conflicting reports over the sequence of events and of how many died in the fighting.

A Pentagon official confirmed an air strike last night during an incursion by insurgents into Afghanistan from Pakistan.

He said two US aircraft chased insurgents and launched an air strike under an agreement that allows coalition forces to enter Pakistan if they are in hot pursuit of a target.

The Afghan interior ministry spokesman, Zemeri Bashary, put the death toll at 31 and insisted most were foreign fighters.

Khalid Farooqi, a politician from Paktika, eastern Afghanistan, said more than nine civilians were also killed. He said the operation apparently targeted the militant commander Mullah Mohammad Nabi and fighters who served under him.

The US-led coalition said four civilians were killed and that "several" militants died in the clashes, in the northern Paktika province. Twelve militants were detained.

According to reports, clashes broke out late last night after Pakistani tribesmen tried to stop Afghan security forces from setting up a mountain-top post.

Damagh Khan Mohmand, a local tribesman who witnessed the fighting, said it lasted four hours. He said Afghan and foreign forces traded fire with Pakistani tribesmen and troops.

Reports suggested the air strike was by a US drone.

Two aircraft bombed several locations, hitting two paramilitary posts in the region, he said.

Maulvi Umar, a spokesman for an umbrella group of Pakistani Taliban, said between 60 and 100 of its fighters attacked Nato and Afghan army troops who had set up bunkers and tents on Pakistani soil.

He claimed up to 40 Afghan troops were killed, several captured and that a Nato helicopter was shot down.

Eight Taliban were killed and nine wounded in the subsequent US air strike. He told Reuters he had heard that US aircraft also bombed a nearby Pakistani post.

Nato in Afghanistan referred inquiries to the US military, whose spokeswoman, Lt Col Rumi Nielson-Green, referred calls to the US embassy in Pakistan. The embassy also declined comment. The Afghan ministry of defence said it had no information on the incident.

Pakistani TV networks reported the incident but gave varying accounts of the death toll and the fighting, which took place in a far-flung border area known as Speena Sooka, or White Peak.

State-run Pakistan television said 18 people died - 10 troops and eight civilians. It said Afghan and foreign forces had tried to set up a military post and were resisted by tribesmen. A Nato air strike then struck a Pakistani military post, PTV said.

Pakistan says it has deployed tens of thousands of troops to police its tribal regions, but Afghan and Nato officials say militants still use the lawless area as a staging post for attacks inside Afghanistan.

The US has in the past used unmanned drones to attack suspected militants inside Pakistan, an ally in its war on terror. Pakistan says the attacks are a violation of its sovereignty.