US President Donald Trump said the US and UK have the “greatest alliance the world has ever known” in a news conference with Theresa May.
The prime minister said they had a “great relationship” that they should “build on”.
There were also frank talks about their differences, including climate change and business with Huawei, she said.
Labour leader Jeremy Corbyn urged Mr Trump to “think on about peace” as he addressed protesters in Westminster.
During a breakfast meeting at St James’s Palace earlier today, Mr Trump said he believed there was scope for further expansion of trade between the US and the UK.
“I think we will have a very, very substantial trade deal,” he told Mrs May.
“This is something you want to do and my folks want to do,” he added. “Stick around. Let’s do this deal.”
BBC assistant political editor Norman Smith said that Mr Trump’s remark to “stick around” was meant almost as a pleasantry rather than a serious plea for Mrs May to reconsider her plan to stand down. “It’s just trying to make this a little bit easier for her,” Mr Smith said.
He added that although Mr Trump “talks a good game” over a trade deal, the UK is likely to be in for a “slow haul”, with most deals taking around 10 years.
The meeting of five British and five American firms, senior ministers and officials was held in a bid to boost trade links.
Mrs May said there were “huge opportunities” for Britain and the US to work together in the future.
“It is a great partnership but I think a partnership we can take even further,” she told the president, adding: “Of course that is with a good bilateral trade deal.”
The US president is also set to hold talks with Conservative leadership hopeful Michael Gove, according to sources close to the environment secretary.
Mr Trump praised Mr Gove’s leadership rival Boris Johnson ahead of his visit to the UK.
A source close to Mr Gove told the Press Association “nothing has been fixed yet”, but added: “Mr Gove was asked last night by Mr Trump’s team if he would be able to meet the president today. He said yes.”
Meanwhile, protesters have gathered in central London to voice their opposition to President Trump’s visit. Demonstrations are also planned in Birmingham, Stoke, Sheffield, Glasgow, Edinburgh, Chester, Leicester, Oxford and Exeter.
Police said more than 3,000 officers have been deployed for the president’s visit.
Mr Corbyn – who boycotted the state dinner – was joined at the rally by members of other political parties, including the Liberal Democrats and the Green Party.
He said he had not refused to meet Mr Trump, saying: “I want to have a dialogue.”
But he criticised the US president’s attack on Mayor of London Sadiq Khan. “I am proud that our city has a Muslim mayor, that we can chase down Islamophobia, anti-Semitism, any form of racism within our society,” he said.
Mr Khan defended his party leader Mr Corbyn’s decision to boycott the state dinner at the palace.
Mr Khan said he felt he and Mr Corbyn shared the view that a “close relationship” with the US president was important.
But he added: “What shouldn’t happen is a state banquet and a state visit with the red carpet rolled out. I think it’s inappropriate. I think those visits should be reserved for leaders who have done something and deserve that. I think it sends the wrong message to be seen to condone some of the things this president has said and done.”
At the protest
By Marie Jackson, BBC News
The whistling and whooping ramps up as Labour leader Jeremy Corbyn takes to the stage in Whitehall. The rainy streets are now packed with cagoule-clad protesters holding aloft umbrellas and placards saying “Dump Trump”.Some have their faces covered with #trumpstinks masks, others wear badges saying “another nasty woman against Trump”.There are mums with small children in buggies who have given up a day in the park to make their young voices heard, alongside seasoned protesters and US expats.A little further up the street, police officers are dealing with a disturbance by the English Defence League.The rain’s holding off for the moment but Mr Corbyn has a message for the visiting president.
“Think on, please, about a world that is aiming for peace and disarmament, that defeats racism and misogyny.” The crowd roars cheers of support.
On Monday Mr Trump was welcomed by the Queen and had lunch at Buckingham Palace with senior royals. He also visited Westminster Abbey and had tea at Clarence House with the Prince of Wales and the Duchess of Cornwall.
He attended a state banquet at Buckingham Palace in the evening, at which Mr Trump praised the “eternal friendship” between the UK and US.
The Queen said the countries were celebrating an alliance which had ensured the “safety and prosperity of both our peoples for decades”.
The president’s visit coincides with the commemorations for the 75th anniversary of the D-Day landings, which the Queen, Mr Trump and other heads of state will attend at Portsmouth on Wednesday.
Analysis: Diplomatic differences
By James Landale, BBC diplomatic correspondent
Donald Trump may deliver his messages hard and from the hip, firing off tweets and juicy quotes.
But the British political establishment is more subtle in the way it sends its diplomatic signals.
The Queen used her banquet speech to remind the president that the international institutions set up after the World War Two were designed to ensure the horrors of conflict were not repeated.
Theresa May presented Mr Trump with a framed copy of the Atlantic Charter, a set of principles centred on freedom and co-operation agreed by Winston Churchill and President Roosevelt in 1941 that formed the basis of many post-war institutions.
Both have been seen as a thinly-veiled message to a president who has repeatedly shown his disdain for multilateral organisations such as the United Nations and Nato.
Equally, the Queen in her own way has placed a small and utterly deniable distance from the president.
Her welcome has been warm and complete but she has not had Mr Trump to stay at Buckingham Palace – too much building work, naturally – nor will she attend his dinner at the US ambassador’s residence – one for Prince Charles, you understand.
The decision by Downing Street to put Eton Mess on the menu for lunch has also been seen by some as a tongue-in-cheek warning for a president who arrived telling newspapers his favourite Old Etonian, Boris Johnson, would be an “excellent choice” to replace Theresa May as prime minister.
All deniable, all open to accusation of over-interpretation. But remember: these state visits are highly choreographed and nothing is said or done without reason.
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