Separately, a NATO service member was killed by insurgents in the country's east, according to a military statement.
No group immediately claimed responsibility for any of the attacks, but Taliban fighters have escalated their activity as US-led foreign forces reduce their presence in the country, having handed over primary responsibility for security to Afghan troops.
Javed Faisal, a spokesman for the provincial governor, initially said the suicide bomber was in a car that was being searched by police, but later said new information indicated the bomber had been on foot. Along with the branch building of the New Kabul Bank, several small shops and vehicles were damaged.
Faisal said at least six people died, four of them civilians, one police officer and one private security guard. Another 24 people were wounded, most of them civilians.
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Taliban spokesmen did not immediately respond to requests for comment. The militant group is especially strong in southern Afghanistan, which is dominated by the ethnic Pashtun community whose members form the bulk of the insurgency in the country.
Zwak said 11 men and one woman died in the attack, and that the vehicle also was hit by several rounds of gunfire.
Such attacks typically target security forces, but, in this case, "the victims are all civilians and had no link with the government," Zwak said.
Afghan and coalition officials have warned that the Taliban would intensify the tempo of their attacks following the Muslim holy month of Ramadan, as they try to take advantage of the two or three months left of good weather before the harsh Afghan winter sets in.
The NATO service member died in "a direct fire attack by enemy forces in eastern Afghanistan" today, according to a statement from the military alliance.
The statement did not give any further details on the person killed. Mostly US troops operate in eastern Afghanistan.