The British troops were attacked near the city of al-Amarah, 200 km north of the British-occupied city of Basra.
Six British personnel were killed in one incident, Prime Minister Tony Blair's spokesman said.
In a second confrontation, British paratroopers on patrol came under fire, the spokesman said. One was wounded and two vehicles were destroyed.
A rapid reaction force that was helicoptered to the scene also came under attack on landing. Seven were wounded, three seriously.
Al-Amarah lies in the predominantly Shia south of the country, which so far has not witnessed the routine resistance activities that have become a commonplace in the mainly Sunni north.
The attacks are also significant because it is the first time British forces have been targeted since US President George W Bush announced an effective end to the war on 1 May 2003.
Meanwhile, US soldiers - who came under resistance attacks in the Iraqi town of al-Ramadi late on Monday - killed five Iraqis in raids on houses.
The raids, described as fierce by Aljazeera's correspondent, took place after a US patrol came under attack.
Two civilians were killed in their homes, including one student who was studying in the garden because of a power cut.
US military sources later said two soldiers had been wounded.
In Fallujah a US tank was destroyed by rocket-propelled grenades. There were no casualties among the US forces who stormed homes, killing one civilian and arresting another.
The man killed was standing at the gate of his home when US troops opened fire, reported an AFP correspondent. He was decapitated by the shots. Occupation troops stood guard by the corpse, preventing citizens from removing it.
US helicopters hovered over the town, a hotbed of resistance attacks.
Searches in Baghdad
US troops also conducted raids early on Tuesday in the Dura area of southern Baghdad.
17-year-old Khaled was arrested
for insulting US troops
A US military official said the searches were being conducted after an attempted hit-and-run on a US soldier at a checkpoint, a few days ago.
American troops detained a 17-year-old student on his way to school for insulting the soldiers and making a gesture imitating shooting.
US Defence official in Washington said US troops clashed with Syrian border guards last Thursday during an attack on a convoy of suspected high-profile members of the former regime.
Another US official, speaking on condition of anonymity, has said the incident may have occured in Syrian territory.
US special forces and airborne troops, backed by aircraft, attacked the convoy which turned out not to have any senior Iraqi officials, said the US official.
US officials said they were treating three Syrian guards.
Syria, whose already cold ties with Washington became more frigid after the war against Iraq, said it had no comment.
Washington indicated there had been no official government contact with Damascus over the incident.
Guards to increase
US military officials said it would
be difficult to guard pipelines
In other developments, a senior oil ministry official said the number of armed guards patrolling the country's fuel pipelines would soon be doubled from 3,000 to 6,000.
The move came after two separate pipeline explosions over the last four days.
An Iraqi oil ministry official, who asked not to be named, said the blasts might have been the result of people trying to extract petrol from the pipelines.
Local residents and Iraqi officials suspected the blasts were due to what they described as "sabotage".
US figures estimate the length of Iraq's fuel pipelines to be more than 7,000 km. US military officials said guarding the key ducts, vital to resuming oil exports, was almost impossible.
The latest blast on Sunday delayed the resumption of exports between the northern Iraqi city of Kirkuk and the Turkish Mediterranean terminal in Ceyhan.