Scotland Yard has revealed that the woman killed in the knife attack in central London was a US citizen and the suspect is a Norwegian national of Somali origin.
The injured were British, American, Israeli and Australian, head of counter-terrorism Assistant Commissioner Mark Rowley confirmed.
Rowley said that so far no evidence has been found of radicalisation that would suggest the man in custody was "in any way motivated by terrorism".
Rowley said specialist detectives from murder and terrorism teams had been working on the case throughout the night.
He went on: "Whilst the investigation is not yet complete, all of the work we have done so far increasingly points to this tragic incident as having been triggered by mental health issues.
"At this time, we believe this was a spontaneous attack and that the victims were selected at random."
Investigators have made "urgent progress" in five strands of work, he said - interviewing the suspect, speaking with his family, witness accounts, address searches and a "full intelligence review" by police and the security service.
On Thursday morning officers searched an address in north London and will search another in south London.
Rowley, Britain's most senior counter-terrorism officer, said: "I emphasise that so far we have found no evidence of radicalisation that would suggest the man in our custody is in any way motivated by terrorism.
"The suspect is a Norwegian national, of Somali origin. I stress though that all the work we have done so far does not suggest that those factors in his background are relevant to the motivation for his actions."
Meanwhile, Metropolitan Police Commissioner Sir Bernard Hogan-Howe has vowed to "protect our capital and those who live, work and visit our city" after the Russell Square stabbings, adding: "A normal night out in our busy capital has ended in horrific circumstances."