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He left his military base in the early hours of the morning and opened fire in at least two homes; women and children were among the dead.
Nato said it was investigating the “deeply regrettable incident”.
Anti-US sentiment is already high in Afghanistan after US soldiers burnt copies of the Koran last month.
US officials have apologized repeatedly for the incident at a NATO base in Kabul, but they failed to quell a series of protests and attacks that killed at least 30 people and six US troops.
Local people have reportedly gathered near the base in Panjwai district to protest about Sunday’s killings, and the US embassy is advising against travel to the area.
Lt Gen Adrian Bradshaw, deputy commander of NATO-led forces, said he was unable to “explain the motivation behind such callous acts”, adding that “our thoughts and prayers are with those caught in this tragedy”.
The soldier has not been named, but is thought to be a staff sergeant.
He is reported to have walked off his base at around 03:00 local time (22:30 GMT Saturday) and headed to nearby villages, moving methodically from house to house.
“Eleven members of my family are dead. They are all dead,” Haji Samad, an elder from Najeeban village, told the AFP news agency.
Haji Sayed Jan, from Alkozai village, was quoted by the AFP as saying: “My home was attacked and I lost four family members”.
A delegation from the provincial governor’s office has arrived in the village to determine exactly what happened, a spokesman said.
The soldier – who had reportedly suffered a breakdown before the attacks – is said to have handed himself over to the US military authorities after carrying out the killings.
The Nato-led International Security Assistance Force (Isaf) said in a statement that US officials in Afghanistan would work with their Afghan counterparts to investigate what happened.
This is the first time Afghan civilians have been targeted by foreign soldiers in this way, the BBC’s Quentin Sommerville reports from Kabul.
However, a US soldier was convicted last year on three counts of premeditated murder after leading a rogue “kill team” in Afghanistan.
Before Sunday’s killings, relations between international forces and the Afghan people were already at an all-time low following the accidental burning of the Koran by US soldiers last month, our correspondent adds.
A senior Afghan intelligence official called the latest attack a “propaganda victory for the Taliban”.
Kandahar is the Taliban’s spiritual heartland and is considered strategically important because of its international airport, its agricultural and industrial output and its position as one of the country’s main trading hubs.
The province has seen heavy fighting between NATO and Taliban forces over the last five years.
Afghan President Hamid Karzai said in a speech on Sunday that his government still expects to sign a strategic partnership with the United States in the next couple of months.
In a televised speech, he said discussions would continue on the precise role the US will play in Afghanistan after NATO hands over security responsibility to Kabul at the end of 2014.
On Friday, Kabul and Washington reached a deal to transfer US-run prisons in the country to Afghan control.