A further 1,325 people have died in the UK within 28 days of a positive Covid test - the biggest figure reported in a single day since the pandemic began.
It means there have been just short of 80,000 deaths by that measure - as another 68,053 new cases were recorded.
Public Health England (PHE) said the number of deaths would "continue to rise until we stop the spread".
It comes as the government launches a new campaign in England urging people to "act like you've got" the virus.
The campaign, including an advert fronted by England's chief medical officer, Prof Chris Whitty, is intended to remind the public Covid is spreading fast, with large numbers showing no symptoms.
In the advert, Prof Whitty says: "Covid-19, especially the new variant, is spreading quickly across the country.
"This puts many people at risk of serious disease and is placing a lot of pressure on our NHS.
"Once more, we must all stay home. If it is essential to go out remember, wash your hands, cover your face indoors and keep your distance from others."
Prime Minister Boris Johnson said: "Our hospitals are under more pressure than at any other time since the start of the pandemic, and infection rates across the entire country continue to soar at an alarming rate."
A message from @CMO_England Professor Chris Whitty on #coronavirus#COVID19 is spreading across the country, putting many at risk and placing pressure on the #NHS— Department of Health and Social Care (@DHSCgovuk) January 8, 2021
As we roll out vaccines, it is vital that we all continue to stay at home to:
▶️ protect the NHS
▶️ save lives pic.twitter.com/uc22i59L0U
Hospital leaders have warned of stretched staffing with 31,624 coronavirus patients in UK hospitals on Wednesday - 46% above the peak during the first wave last year.
Dr Ian Higginson, vice president of Royal College of Emergency Medicine, said the situation in London and south-east England was "pretty dire" and would get worse in the rest of the country before long.
"We're heading for some really dark times, I fear, in this phase of the pandemic," he said.
Richard Mitchell, chief executive of Sherwood Forest Hospitals NHS Trust, said the increase in patients seen in London was now affecting his area in Nottinghamshire.
He said: "Critical care is exceptionally busy and the colleagues who work here are tired, they're fatigued and they're worn out."
Meanwhile, a third Covid vaccine received emergency approval for use in the UK with 17 million doses of the jab, made by US firm Moderna, pre-ordered by the UK.
The vaccine joins the Pfizer-BioNTech and Oxford-AstraZeneca jabs in being approved, with close to 1.5 million people now vaccinated in the UK.
Dr William Welfare, Covid-19 response director at PHE, said: "Each life lost to this virus is a tragedy, but sadly we can expect the death toll to continue to rise until we stop the spread.
"Approximately one in three people who have coronavirus have no symptoms and could be spreading it without realising it.
"To protect our loved ones it is essential we all stay at home where possible. This will reduce new infections, ease the pressure on the NHS and save lives."
London Mayor Sadiq Khan said the spread of Covid in the capital was now "out of control", as he declared a "major incident".
This means the emergency services and hospitals cannot guarantee their normal level of response, and allows special arrangements to be implemented.
The previous highest daily death toll - 1,224 - was recorded on 21 April 2020 during the UK's first lockdown. Daily deaths were in the single figures as recently as September.
The UK has recorded the fifth-highest number of deaths behind the United States, Brazil, India and Mexico, according to Johns Hopkins University.
Tragic numbers will continue
We are now seeing the record numbers of cases over the Christmas period translate into record numbers of deaths.
And with new infections rising rapidly - more than 1.1 million people in England estimated to be infected with Covid-19 last week - these tragic numbers are set to continue for some time.
And that is mainly because of the new variant form of the virus which is thought to be between 30-70% more transmissible.
The administration of the vaccines to at-risk groups should see a reduction in the numbers dying by the end of the month and the numbers having to go into hospital going down sometime after that.
That is the other way around from what you normally hear - but that it because a successful vaccine programme will initially remove those most likely to die from the path of the virus.
Fitter or younger people - who are less likely to die but could still end up occupying hospital beds - won't be getting their jabs for some time yet.
The advent of spring's better weather should also help cases to fall, but ministers will have to decide what level of risk - and deaths - society is prepared to tolerate.
Friday saw 619,941 tests conducted in the 24 hours to 09:00 GMT - also a new record.
England, much of Scotland, Wales and Northern Ireland continue to be under strict national measures, with stay-at-home orders in place for most people.
The R number - the rate at which an infected person passes on the virus to someone else - is now estimated to be between 1.0 to 1.4, meaning the epidemic is growing between 0% and 6% per day.
Covid infections rose by almost a third between Boxing Day and 3 January, reaching 70,000 new cases a day according to a major study.
In a different piece of research, an estimated 1.2 million people in total had Covid over a similar time period, the Office for National Statistics said.
Boris Johnson pledged on Thursday to use England's lockdown to implement an "unprecedented national effort" to offer vaccination to those at the highest risk from Covid by 15 February.
He said the Army would be drafted in to use "battle preparation techniques" to achieve the goal, which could see up to 15 million people offered a vaccine by the middle of next month.
In another development, from next week all travellers to the UK will need to show a recent negative test result before they arrive.
Latest data in three graphs
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