Boris Johnson has warned EU leaders not to give the UK “punishment beatings” for Brexit “in the manner of some World War Two movie”.
The foreign secretary said penalising “escape” was “not in the interests of our friends and our partners”.
PM Theresa May set out her Brexit strategy, including leaving the EU single market, in a speech on Tuesday.
European Commission President Jean-Claude Juncker promised to work for “good results” from Brexit talks.
And, at Prime Minister’s Questions, Mrs May clashed with Labour leader Jeremy Corbyn, telling him she had a “plan” and he did not “have a clue”.
Mr Corbyn accused her of “threatening to turn Britain into an offshore tax haven”.
- Brexit Live: Rolling reaction to May’s speech
- The UK’s Brexit plans: What we now know
- Theresa May quotes on Brexit: Then and now
- Europe sees UK set for ‘hard’ Brexit
- Reality Check: How could customs union deal work?
With just over two months to go before the UK government is due to get Brexit talks under way, Mr Johnson was asked on a trip to India about comments by an aide to French President Francois Hollande, who said the UK should not expect a better trading relationship with the EU after leaving it.
He replied: “If Monsieur Hollande wants to administer punishment beatings to anybody who chooses to escape, rather in the manner of some World War Two movie, then I don’t think that is the way forward.
“I think, actually, it’s not in the interests of our friends and our partners.”
Downing Street later said Mr Johnson “was not in any way suggesting anyone was a Nazi”.
The spokeswoman said the remarks were “all being hyped up” and that Mr Johnson was making a point about a punitive approach, using a “theatrical comparison”.
Questioned whether there was concern about raising the issue of World War Two in this context she said: “There is not a government policy of not talking about the war.”
A Labour spokesman said: “The foreign secretary has a habit of making wild and inappropriate comments. Talking about World War Two in that context is another one of those and not something that’s going to improve the climate for negotiations.”
Mr Johnson, a key Brexit campaigner, earlier wrote in the Daily Telegraph that the UK would “no longer have our trade policy run by the EU Commission”.
He added: “That means – crucially – that we will be able to do new free trade deals with countries around the world. They are already queuing up.”
EU leaders have begun to deliver their verdicts on Mrs May’s speech, in which she also warned against trying to “punish” the UK for Brexit and hinted she could walk away from talks if not happy, stating that “no deal for Britain is better than a bad deal for Britain”.
Mr Juncker said he would work to ensure Brexit talks are carried out “according to the rules and they yield good results”.
He added: “I welcome the clarifications given by Mrs May, but I said to her last night that a speech will not launch the negotiations.”
Maltese Prime Minister Joseph Muscat, whose country currently has presidency of the EU, warned: “We want a fair deal for the United Kingdom, but that deal necessarily needs to be inferior to membership.
“Thinking it can be otherwise would indicate a detachment from reality.”
The Czech Republic’s Secretary of State for EU Affairs, Tomas Prouza, told BBC Radio 4’s Today programme it was important that a Brexit deal “makes sense for both sides”.
What the newspapers say about May’s speech
In its headline, the Times sums up the prime minister’s message to the EU as “Give us a fair deal or you’ll be crushed”.
Meanwhile, the Brexit-supporting Daily Mail draws parallels with Margaret Thatcher, saying Mrs May exhibited the “steel of the new Iron Lady”.
The Guardian, which opposed Brexit in the referendum, found the speech a “doubly depressing event” – a reality check for those who want to keep the UK in the single market while being riddled with its own streak of “global fantasy”.
The Financial Times praises the prime minister’s “bold vision” but warns that the road ahead will be perilous.
The Sun’s front page is mocked up as a Biblical tablet of stone bearing the single-word headline “Brexodus”.
Downing Street said European leaders spoken to by Mrs May in a series of phone calls had welcomed the “clarity” of her plans.
In her speech, the prime minister suggested the UK could cut its corporate tax rates to compete with the EU if denied access to the single market.
And she promised that Parliament would get to vote on the final Brexit deal.
Asked what would happen if MPs and peers rejected it, Brexit Secretary David Davis told Today: “They won’t vote it down. This negotiation will succeed. It will succeed.”
Analysis – By Laura Kuenssberg, BBC political editor
Since the referendum Theresa May and her ministers have simply refused to be so explicit.
For months, some ministers have privately whispered about complex solutions that might keep elements of membership – the choices not being binary, mechanisms that might give a sort of membership with a different name.
Well, no more. The simple and clear message from Theresa May’s speech this morning is that we are out.
The government says it will invoke Article 50 of the Lisbon Treaty, getting formal Brexit talks with the EU under way, by the end of March, with discussions set to last up to two years after that.
At Prime Minister’s Questions, Mr Corbyn urged Mrs May to “stop her threat of a bargain basement Brexit, a low-paid tax haven on the shores of Europe”.
He added: “It won’t necessarily damage the EU but it would certainly damage this country.”
Mrs May told MPs: “What I set out yesterday was a plan for a global Britain, bringing prosperity to this country and jobs to people and spreading economic growth across this country.”
European press reaction to Brexit speech
A “catalogue of demands with some threats thrown in” is German news magazine Der Spiegel’s description of Theresa May’s Brexit speech. It says that her desire to leave the single market while retaining access to trade with Europe shows that her government is “not just nasty but also blind to reality”.
Germany’s Die Welt also mocks her with the headline “Little Britain” and accuses her of leading the country into “isolation”.
In Italy, La Repubblica’s front page reads “Brexit: London raises its wall ‘away from the EU and the single market'”.
France’s Liberation remarks that Mrs May’s comment that no deal is better than a bad one suggests that she is threatening to turn Britain into a tax haven. “If this is not blackmail, it looks a lot like it,” it says.
Liberal Democrat leader Tim Farron said: “Ripping us out of the single market was not something proposed to the British people. This is a theft of democracy.”
UKIP leader Paul Nuttall said he feared a “slow-motion Brexit”, adding: “We want this done quickly.”
Scottish First Minister Nicola Sturgeon claimed leaving the single market would be “economically catastrophic”.
She hinted at a second independence referendum, saying Scotland – which voted against Brexit – should have “the ability to choose between that and a different future”.