The 52-year-old suspect, who lived close to the scene where Cox was killed, is described by his neighbours as a quiet loner with a passion for gardening.
The UK has suspended campaigning ahead of next week's EU membership referendum after a British lawmaker was murdered on Thursday in what appears to be a politically-motivated assassination.
Jo Cox, a member of parliament for the centre-left opposition Labour Party, was stabbed and shot on the street as she was preparing to meet supporters in her constituency in the Birstall district of Leeds, northern England.
The 41-year-old mother of two was reportedly killed by a male suspect in his 50s who shouted "Britain first" before carrying out the attack. The suspect was later arrested nearby, and was found to be carrying weapons, West Yorkshire regional police said.
Britain First is the name of a far-right movement which describes itself as "a patriotic political party and street defence organisation" on its official website.
But the group's deputy leader, Jayda Fransen, was quick to distance Britain First from the murder, calling it "absolutely disgusting."
Cox was an avid campaigner for Britain's EU membership. She is the first British MP to be killed while serving in office since Ian Gow was assassinated in a car bombing by the Irish Republican Army in 1990.
Nevertheless, both the Remain and Leave camps announced they would be suspending their campaigns on Friday.
Temporary Chief Constable Dee Collins told reporters that the police are not currently in a position to discuss any motive for her murder.
A "very significant investigation with large numbers of witnesses" was under way, Collins said, adding that police were not looking for anyone else in connection with the incident.
West Yorkshire police called the attack "a localised incident, albeit one that has a much wider impact."
Local sources named the suspect as Tommy Mair.
The 52-year-old suspect, who lived close to the scene where Cox was killed, was described by neighbours as a quiet loner with a passion for gardening. His family said he had a history of mental illness.
Next-door neighbour Diana Peters, 65, told Reuters that she had known Mair since he was a boy and he never had visitors.
"I'm totally devastated - I didn't want to believe it. He's been very helpful to me. Anything I asked him to do he did very willingly and sometimes without my needing to ask," she said.
Mair had taught English to foreigners in the local community for several years and was brought up by his grandparents, she said. His mother is now in a local care home, she added.
Speaking after her death, her husband Brendan said "she would have wanted two things above all else to happen now: one, that our precious children are bathed in love and two, that we all unite to fight against the hatred that killed her."
British Prime Minister David Cameron said the killing was a tragedy. "We have lost a great star," he said. "She was a great campaigning MP with huge compassion, with a big heart. It is dreadful, dreadful news."
The death of Jo Cox is a tragedy. She was a committed and caring MP. My thoughts are with her husband Brendan and her two young children.— David Cameron (@David_Cameron) June 16, 2016
It's right that all campaigning has been stopped after the terrible attack on Jo Cox. I won't go ahead with tonight's rally in Gibraltar.— David Cameron (@David_Cameron) June 16, 2016
Cameron, who leads the governing Conservative Party, also said he was cancelling his planned trip to Gibraltar, a British peninsula on the southern Iberian coast.
Labour Party leader Jeremy Corbyn paid tribute to his fallen colleague in a televised statement.
"We've lost a wonderful woman, we've lost a wonderful member of parliament, but our democracy will go on," he said. "As we mourn her memory, we'll work in her memory to achieve that better world she spent her life trying to achieve."
The whole of the Labour family, and indeed the whole country, is in shock and grief at the horrific murder of Jo Cox pic.twitter.com/obic5pOCS3— Jeremy Corbyn (@jeremycorbyn) June 16, 2016
Her fellow Labour MP Sarah Champion also expressed her grief, saying "[Cox] fights so hard for the things she believes in. I cannot believe anyone would do this to her."
Cox, a Cambridge University graduate, was well-known for her humanitarian work, particularly regarding women's issues, having worked for Oxfam for a decade.
"Oxfam is deeply shocked to hear the news. Our thoughts and sympathies are with Jo and her family at this difficult time," Oxfam's chief executive Mark Goldring said.
Leeds North East MP Fabian Hamilton said he was "completely devastated" by the killing, adding that everyone is "totally shocked."
Huddersfield MP Barry Sheerman, who mentored Cox, said she was "one of the real talents in parliament, people loved her in the constituency."
Hundreds of people attended a vigil at a local church in the Birstall constituency while the British flag was lowered to half-mast above the Houses of Parliament in London.
A Buckingham Palace spokesman said Queen Elizabeth II will write a private letter of condolences to her husband.
Meanwhile, US Secretary of State John Kerry described the murder as "an assault on everybody who cares about and has faith in democracy."