Other countries are “queuing up” to sign trade deals with the UK once it leaves the EU, Boris Johnson has said.
The foreign secretary also said the UK would not be “hauling up the drawbridge” despite new migration controls promised by Theresa May.
Mrs May set out her Brexit strategy – which includes pulling out of the EU single market – in a speech on Tuesday.
She also warned the EU not to try to “punish” the UK, saying she would walk away unless the right deal was offered.
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With just over two months to go before the government is due to get official Brexit talks under way, Mr Johnson wrote in the Daily Telegraph: “We will no longer be part of the common commercial policy, or bound by the Common External Tariff, and we will no longer have our trade policy run by the EU commission.
“That means – crucially – that we will be able to do new free trade deals with countries around the world. They are already queuing up.
“Under EU rules, we are not formally allowed to negotiate these new treaties until we leave. But there is nothing to say that ideas cannot be pencilled in.”
Mr Johnson also said the UK would “continue to share European values”.
“We will continue the joyous exploration of other European culture and civilisation that has been expanding ever since the dawn of cheap air travel, and we will continue to welcome vast numbers of EU tourists to the UK.
“We are not slamming the door to migrants, or hauling up the drawbridge.”
EU leaders are set to give their verdicts on Mrs May’s speech in the next few hours.
But Downing Street said they had already welcomed the “clarity” of her plans during a series of private phone calls.
In her speech, Mrs May said it was not her intention to “undermine” the EU or the single market, but she warned against a “punitive” reaction to Brexit.
She suggested the UK could cut its corporate tax rates to compete with the EU if denied access to the single market, and added that “no deal for Britain is better than a bad deal for Britain”.
Speaking on BBC Newsnight, Labour leader Jeremy Corbyn said Mrs May’s “implied threat” to turn the UK into a “low-corporation tax haven” was an “odd way of approaching a constructive relationship with a whole continent”.
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.
But Brexit Secretary David Davis told BBC Radio 4’s Today programme the response from Brussels had been “positive” and the prime minister’s comments had been what “they were waiting for, frankly”.
He added that a trade deal with the EU should be reached within “a year or two” of Brexit negotiations ending.
Mr Davis said: “My job is to deliver the best deal possible… They want us to get on with it. The simple truth is we need to do this expeditiously.”
The prime minister promised in her speech that Parliament would get to vote on the final Brexit deal.
Asked what would happen if MPs and peers rejected it, Mr Davis responded: “They won’t vote it down. This negotiation will succeed. It will succeed.”
The government says it will invoke Article 50 of the Lisbon Treaty, getting official Brexit talks under way, by the end of March. Discussions are set to last up to two years.
Shadow Brexit secretary Keir Starmer told BBC Breakfast there was “no mandate” for Mrs May’s “bargain basement tax haven threat”, which he said would make people poorer and be “totally inconsistent” with the protection of “workers’ rights and a fairer Britain”.
But he added: “What she did say, though, which was important, is that she intends to have as one of her objectives tariff-free access to the single market… that is really important for business.”
The Czech Republic’s Secretary of State for EU Affairs, Tomas Prouza, told Today it was important that a Brexit deal “makes sense for both sides”.
He also said it was important for Mrs May to clarify what will happen after the UK leaves the jurisdiction of the European Court of Justice.
Guy Verhofstadt, the European Parliament’s chief Brexit negotiator, tweeted: “Threatening to turn the UK into a deregulated tax haven will not only hurt British people – it is a counter-productive negotiating tactic.”
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”.