Health Secretary Matt Hancock said he "understands how difficult" coronavirus is for the country, adding: "I've lost two people I was fond of".
The cabinet member, who has himself recovered from Covid-19, delivered the government's daily briefing from Downing Street today, and the emotion was clear in his voice as he spoke of the losses.
He said he "understands how difficult" coronavirus is for the country - but called on the "small minority" who are not following social distancing rules to stop risking their own lives and those of others.
He also offered his "profound sympathies" to the families and friends of those who have died.
He added: "I've lost two people that I was fond of so I understand what a difficult time this is for the country."
Although he did not identify the people he was referring to, it is thought one of them may have been Steven Dick, who served as Deputy British Ambassador to Hungary.
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He died on March 24 aged just 37.
Following his death, Mr Hancock wrote on Twitter: "We worked together when I was Culture Secretary. He exemplified everything best in the civil service: dedicated, creative, & thoughtful. Such a sad loss, too soon. My heart goes out to his family and friends."
Mr Hancock said no changes for social distancing rules were "imminent", but the government would not rule out further steps.
He added: "Following these rules is mission-critical if we're to protect the NHS, slow the spread and give the NHS time to expand capacity and so save lives.
"The more people follow the rules then the faster we will all be through this.
"So I say this to the small minority of people who are breaking the rules or pushing the boundaries: you're risking your own life and the lives of others and you're making it harder for us all."
In the same press conference, Dr Jenny Harries, Deputy Chief Medical Officer for England, warned against "lifting the lid" on the UK's lockdown too early - as this could lead to a "second spike".
Mr Hancock said coronavirus "thrives on human contact" and urged Brits to protect themselves, their loved ones and the NHS by staying at home.
He also paid tribute to NHS workers, saying he had been "humbled" watching them in action.
The Health Secretary said 300 ventilators had arrived from China, and said development in the UK was moving "at pace".
It comes as the UK's death toll of those diagnosed with coronavirus rose by 621 to 4,934 - a slight decrease from yesterday.
Mr Hancock said the evidence shows "a very high proportion of the public" understand the Government rules and are following them, but he added: "We need everybody to be following those rules."
When asked if people are safe to visit family if they have been following the social distancing guidelines correctly, Mr Hancock said "uncertainty" over how the disease is spread means they should not.
He added: "This virus can spread in all sorts of different ways, both from directly respiratory and also through objects and through touching objects that others then touch - the uncertainty over who has it, especially before they have symptoms, means that the only effective way of bending the curve down is to reduce that social contact."
He urged people suffering coronavirus symptoms to report their experiences online in an effort to better understand the killer bug.
Mr Hancock said the NHS has launched a symptom-checking service on its website.
He said: "This weekend we have launched our latest technology, a new coronavirus status checker.
"It asks people who currently have coronavirus symptoms to share their experiences to help us in our fight against the virus and it will help make use of this information which will, in turn, help us to understand the disease."
He said the website can be found at www.nhs.uk/coronavirus-status-checker.
Dr Harries emphasised "caution" is needed on the number of lab tested cases.
She said: "You will be aware that our capacity is increasing all the time on laboratory tests. So we need to interpret this with a little bit of caution.
"We are following to a certain degree an epidemic curve and we can see the rise and we will of course in due course be looking to see that level off, but you will notice that over the last 24 hours or so there is an apparent big surge there.
"We just need to keep in mind that we need to watch for a trend over time because as our testing capacity increases, inevitably we will find more cases."
Dr Harries added the data showing the number of hospital admissions for Covid-19 gave an indication of how hard hospitals are working across different parts of the country.
Dr Harries said: "Clearly London, we all know, has been working super-hard on the front line and those cases continue to rise, now with an increase in the last 24 hours in the north west.
"We still, as the secretary of state has said, have capacity in all of our hospitals to manage these cases and we're hoping in due course if everybody continues to practice social distancing that those cases will start to fall.
"Clearly there is a delay in the number of hospital admissions from when people often contract the disease and then become more ill and need more intensive care, so there will be a lag on some of this data compared with our social distancing measures."
Dr Harries said it is important to "continue to look and where possible learn" from data from other countries.
Earlier Mr Hancock said that outdoor exercise could be banned completely if people do not stick to social distancing rules.
He said sunbathing in public spaces was against the Government guidance - and told those who disobeyed are putting their own and others' lives at risk.
NHS England gave the breakdown by region of the 555 deaths of patients with Covid-19 as:
- East of England 40
- London 174
- Midlands 74
- North East & Yorkshire 103
- North West 47
- South East 81
- South West 36
The number of people who have died in Northern Ireland after contracting coronavirus has risen by seven to 63, health officials said. Testing there has resulted in 91 new positive cases, bringing the total number of confirmed cases in the region to 1,089.
Scottish First Minister Nicola Sturgeon said two more people have died after testing positive for coronavirus in Scotland, bringing the total deaths there to 220.
And Public Health Wales said 12 more deaths of people who had tested positive for Covid-19 have been reported, taking the number of deaths in Wales to 166.
