The chief constable of the Devon and Cornwall force, Shaun Sawyer, said Jake Davison, 22, had a firearms licence in 2020 but it was not clear whether it related to the weapon he used in his attack.
Officers have been told the weapon was a “pump-action shotgun” and although a weapon has been retrieved they are yet to confirm if that was the case, Sawyer said.
At a press conference in Plymouth, Sawyer said Davison murdered a 51-year-old woman known to him at an unidentified address in Biddick Drive on Thursday night before leaving the property and shooting dead the three-year-old girl and a 43-year-old male relative of the girl.
Davison then shot two other residents, a man and woman aged 33 and 53 respectively, in Biddick Drive, who are in hospital with significant but not life-threatening injuries.
He then entered adjacent parkland where he shot a 59-year-old man, who died at the scene, Sawyer said. Then, in Henderson Place, he shot a 66-year-old woman who later died at Derriford hospital in Plymouth.
It took six minutes for the police to arrive on the scene, he said. Sawyer said terrorism was not the focus, but they still had an “open mind”.
In videos posted to social media, Davison discussed being “beaten down” and “defeated by life”. Under the name Professor Waffle, he said he did not have “any willpower to do anything any more”, in a recording dated 28 July, and discussed the misogynistic “incel” movement.
Although Davison said he did not “clarify” himself as an “incel” – the abbreviation used online for “involuntarily celibate”, for those unable to find a romantic or sexual partner despite desiring one – he spoke about “people similar to me have had nothing but themselves”.
Other social media posts from about 2018 also suggest he was a fan of the former US president Donald Trump and a supporter of the UK Libertarian party.
The prime minister, Boris Johnson, tweeted: “My thoughts are with the friends and family of those who lost their lives and with all those affected by the tragic incident in Plymouth last night. I thank the emergency services for their response.”