A man shot dead five people, including a 3-year-old girl, in violence that police in the southern English city of Plymouth believe began with a domestic dispute and the killing of his mother.

Mass shootings are rare in the United Kingdom, where gun ownership is relatively low, and Thursday's six-minute rampage was the worst such incident in more than decade.

Police on Friday named the shooter as Jake Davison, a 22-year-old crane operator. He turned his gun on himself after killing the five victims on Thursday evening, the police said.

Devon and Cornwall Police Chief Constable Shaun Sawyer said police had found no motive but they were not considering terrorism or any far-right associations, although they were trawling through Davison's computer.

“We believe we have an incident that is domestically related, that has spilt into the street and seen several people within Plymouth losing their lives in an extraordinarily tragic circumstance,” Sawyer told reporters.

The shooting started at about 6pm on Thursday, with the offender first killing his 51-year-old mother Maxine Davison.

He then ran outside and immediately shot dead the young girl, named by police as Sophie Martyn, in the street along with her father Lee Martyn, 43.

Davison shot at two other passers-by who were badly injured, then entered a park and shot dead another man, Stephen Washington, 59. He also shot Kate Shepherd, 66, who died later in hospital.

Davison then turned the gun on himself before firearms officers could tackle him. The deadly shooting spree was over in just a few minutes.

Sawyer said witnesses described the weapon as a pump-action shotgun. He could not say whether Davison had mental health issues.

Britain's police watchdog, the Independent Office for Police Conduct, said it will investigate the police decision to return a shotgun and license for it to Davison last month after taking them from him eight months ago because of an assault allegation.

In videos posted on the internet, Davison had complained of not losing his virginity as a teenager and described himself as an “incel” — or involuntary celibate. He complained in the videos of being beaten down by life, the Times reported.

Britain has suffered a number of deadly militant attacks in the past several years, but this was the worst mass killing of its kind since a taxi-driver killed 12 people then shot himself in a rampage in Cumbria, northern England, in June 2010.

The deadliest mass shooting in Britain’s modern history is the 1996 massacre in Dunblane, Scotland, when a gunman killed 16 pupils and a teacher at the local school before killing himself.