Other soldiers were wounded in the attack, officials said.[break]
The killer was still on the run, but "strenuous efforts" were being made to find him, they said, voicing determination the deadly assault would not damage trust between foreign forces and the Afghan counterparts they are training.
A spokesman for Task Force Helmand, Lieutenant Colonel James Carr-Smith, said the three soldiers from the 1st Battalion The Royal Gurkha Rifles Battle Group, from Nepal, were killed in southern Nahr-e Saraj, Helmand province.
"We believe these were the actions of a lone individual who has betrayed his International Security Assistance Force (ISAF) and Afghan comrades," he said.
British Prime Minister David Cameron vowed British troops would not change the way they work with the Afghan military despite the killings.
"I think it´s absolutely essential that we don´t let this appalling incident change our strategy or our approach," Cameron said, calling for a "proper investigation" into what had happened.
"The right thing for us to do is to keep with our strategy of working with and building up the Afghan National Army (ANA)... it´s when that happens that we will be able to bring our troops back home."
Cameron also described the killer as a "rogue element" within the ANA.
Britain has around 10,000 soldiers in Afghanistan as part of an international force fighting the Taliban. Cameron has signalled he would like to see British combat troops withdraw in five years´ time.
Britain´s Defence Secretary Liam Fox said: "Training and developing the Afghan National Security Forces is vital to the international mission in Afghanistan and today´s events will not undermine the real progress we continue to make.
"British and ISAF forces are working shoulder to shoulder with Afghans and will continue to do so undeterred."
Deputy ISAF commander Lieutenant General Nick Parker described the incident, which left a number of other soldiers wounded, as "a really serious breach of trust" that would have an impact on those on the base.
Britain´s Ministry of Defence said that the "soldiers were killed in a suspected premeditated attack by a member of the Afghan National Army using a combination of weapons".
The SITE monitoring service reported that the Taliban claimed an Afghan soldier had killed eight British soldiers in Nahr-e Saraj district.
The soldier had opened fire on sleeping troops at a base, also wounding four, according to a statement quoting Taliban spokesman Muhammad Yusuf and posted on the Afghan Taliban website and jihadist forums, SITE said.
The soldier was taken to a "safe place" after surrendering himself to the "mujahideen", according to the Taliban, who are known to exaggerate claims.
The killings follow a similar attack in November, when an Afghan policeman shot dead five British soldiers at a checkpoint in southern Helmand province, where most of Britain´s troops are based.
The Afghan presidency called for a thorough investigation into the latest incident and expressed regret at a news conference but was unable to confirm any details of the incident.
US General David Petraeus, who assumed command of NATO troops in Afghanistan earlier this month, also offered his condolences.
"We have sacrificed greatly together and we must ensure that the trust between our forces (Afghan and international) remains solid in order to defeat our common enemies," said Petraeus.
Helmand is one of the most volatile regions of southern Afghanistan, where about 1,000 British troops are expected to leave Sangin, making way for US forces, and be redeployed to central Helmand by the end of the year.
The deaths bring to 356 the number of foreign troops to have died in the Afghan war so far this year, according to an AFP tally based on a count kept by icasualties.org. The total for last year was 520.
The United States and NATO have over 140,000 troops in Afghanistan fighting a Taliban insurgency, with the number due to rise to 150,000 in coming weeks.
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