The blast occurred on Sunday near the three-storey building in central Basra at about 7.30am (0330 GMT). Those who escaped injury included former Basra governor Hasan al-Rashid, a senior local leader of the brigade, said police Captain Mushtaq Kadhim.
Reuters reported police and witnesses as saying a child was killed in the blast.
The Badr Brigade is the military wing of the Supreme Council of Islamic Revolution in Iraq (Sciri), the largest Shia party in the Iraqi government.
It was not immediately known who carried out the attack in Basra, a mostly Shia city in southern Iraq where the majority of Britain's 8500 soldiers are based.
Despite earlier claims that British soldiers had created better security conditions in southern Iraq than other areas of the country beset by violence, Shia militias appear to have been growing in power in the region, infiltrating police forces and local political organisations, and allegedly attacking British and US forces.
The Guardian newspaper quoted Basra's police chief Hasan al-Sadi in May as saying that the militias had become the "real power" in Basra and that he trusted only 25% of his own police force.
There is fighting between the
Badr Brigade and other groups
Al-Qaida in Iraq, the armed group purportedly led by Jordanian-born Abu Musab al-Zarqawi, last month purportedly threatened in an audiotape to unveil a new unit to eradicate the Badr Brigade.
But fighting also has occurred in that region between the Badr Brigade and other Shia militias, including the al-Mahdi Army militia, which is associated with the Shia leader Muqtada al-Sadr.
Another Shia power is the Fadila Party, which won tenuous control of the provincial government this year from Sciri after allying with smaller parties.
British forces criticised
Recently, Basra province governor Muhammad al-Waili, a Fadila member, sharply criticised British forces.
On Saturday, al-Waili said British forces were compromising security in the region by conducting raids and arrests without coordinating them with Iraqi security forces.
Blair's government suspects Shia
groups are receiving foreign help
On Thursday, British forces detained 12 Iraqis in a raid on a home in Basra, accusing them of being members or supporters of the al-Mahdi Army, which is suspected of carrying out recent attacks on British and US troops in the region with help from neighbouring Iran.
Last week, British Prime Minister Tony Blair said his government suspected that Iran and Lebanon's Hizb Allah might be supplying technology and explosives to Shia groups operating in Iraq, but he provided no proof.
US marine killed
A US marine was killed by a roadside bomb in the western town of Ramadi, the military said in a statement on Sunday.
The soldier died "while conducting combat operations against the enemy when the vehicle he was in was attacked with an improvised explosive device", the statement said.
The death brought to at least 1954 the number of US military personnel killed in Iraq since the US-led invasion of March 2003, according to Pentagon figures.