The row came as the Prime Minister’s historic Lancaster House speech was welcomed by Leave campaigners and business groups.
British Retail Consortium chief executive Helen Dickinson described it as “an ambitious plan with the right priorities.”
Richard Tice, co-chairman of Leave Means Leave was delighted that Mrs May had taken up his organisation’s main message that “no deal is better than a bad deal.”
He said: “We welcome her commitment to delivering the democratic will of the people and the tough position she has set out ahead of negotiations with the EU.
“We share her vision of an ‘independent, self-governing Global Britain’ outside of the single market, without the restrictive elements of the customs union and no longer ruled by the European Court of Justice.”
Lord Green of Deddington, chairman of Migration Watch UK, said: “These are clear decisions and they are the right ones.
“The government must preserve skilled migration in both directions while achieving a sharp reduction in migration from the EU for low skilled work.”
Leave.EU founder Arron Banks warned that Mrs May needs to live up to her words and warned of a “Remainer coup” when the deal is put before parliament.
The billionaire businessman said: “With the promise of a parliamentary vote on her deal with the EU, May has opened the door to a Remainer coup. The House of Lords is a chamber of party political cronies, unfit for purpose.”
Mr Banks’ warning followed a briefing to journalists by Mr Farron who made it clear that he wants a second referendum and an option to stay in the EU.
Leading an embittered Remoaner backlash, the Lib Dem leader said: “By conceding the half elected Parliament will have a vote she [the Prime Minister] has effectively conceded that Article 50 is revocable.
He went on: “We will argue that the best position of the British government is to make sure that your option is not a bad deal or falling out into outer darkness.
“No Prime Minister who gave a monkey’s about the British people would leave those two options.
“One option would bound to be the retention of membership of the European Union.”
Despite his desire to overturn last year’s referendum result, Mr Farron ironically described Mrs May’s announcement that she will honour the will of the British people and not have a second referendum as “a theft of democracy”.
Leading Brexit campaigner and MEP David Campbell Bannerman said: “He [Farron] accused the Prime Minister of a ‘theft of democracy’ but the real theft of democracy is the Liberal Democrats claiming they are ‘democrats’ when they do everything possible to prevent the result of a fair and free referendum.
“Tim Farron and the other eight hapless Liberal anti-Democrats should wake up and get behind the huge benefits Brexit offers rather than continuing to fight the corner of the unelected EU bureaucrats.”
Outlining Mrs May’s speech to MPs in Parliament, Brexit Secretary David Davis said the Brexit strategy, and the proposals are a “responsible, thoughtful but realistic plan”.
Shadow Brexit secretary Sir Keir Starmer said Mrs May’s position meant she had ruled out a “hard Brexit” with no agreement and World Trade Organisation rules governing future dealings with the EU and promised his party will not seek to block Article 50.
There was also praise for the Prime Minister from leading Tory Remain MP Anna Soubry.
She said: “Can I make it very clear that I welcome her – I nearly said Her Majesty – the Prime Minister’s speech and the statement made by (Mr Davis).
“I think it is realistic, I think it provides much-needed clarity and I think the tone is to be hugely welcomed.”
However, other Remoaner MPs were intent on continuing Project Fear.
Labour MP Chuka Umunna, of the Vote Leave Watch campaign, said: “Now the Prime Minister is giving up single market membership before negotiations have even begun, and has not ruled out the nightmare scenario of no deal being done, which would mean Britain falling on to WTO (World Trade Organisation) rules in our trade with the EU.
“This would increase tariffs, cut jobs, and lead to higher inflation.”
CBI director-general Carolyn Fairbairn, who was also part of the Remain Project Fear campaign, also attacked leaving the single market
She said: “Ruling out membership of the single market has reduced options for maintaining a barrier-free trading relationship between the UK and the EU. But businesses will welcome the greater clarity and the ambition to create a more prosperous, open and global Britain, with the freest possible trade between the UK and the EU.”
But the Institute of Directors (IoD) backed Mrs May’s vision and clarity.
Allie Renison of the IoD, said: “We welcome the level of detail provided in the PM’s speech and her commitment to providing certainty wherever possible, which is absolutely vital for business if they are to navigate and make the best of Brexit.”
January 17th, 2017: Express