Her government on Monday avoided a humiliating defeat in Parliament when it narrowly won another vote over her Brexit customs bill - but only after reluctantly accepting amendments put forward by Brexit hardliners.
But former transport minister Mr Hammond, who joined five other Tory backbenchers in tabling the amendment, insisted its provisions were "entirely in line" with Mrs May's Brexit White Paper. The amendment was passed by 305 votes to 201, which forced May to maintain as an important goal the UK's continued adherence to the European regulatory network for medicines.
The concessions left Tory Remainers furious, while three pro-Brexit Labour MPs who voted with the government, and Lib Dems Sir Vince Cable and Tim Farron who were both absent, were criticised within their own parties.
Despite these warnings, 12 Conservative MPs voted against the government.
However, the government narrowly lost (305-to-301) a vote on the medicines regulatory regime, which means the United Kingdom will seek associate membership of the European Medicines Agency (EMA) and will in any event follow European regulation. The bill now moves on to the House of Lords.
May's challenges came as some British politicians on Tuesday again questioned the legitimacy of the Brexit vote after the electoral watchdog said the official group campaigning for Britain's exit in the 2016 referendum broke electoral laws.
Sir Geoffrey Howe's 1990 speech after resigning as deputy prime minister over differences with Margaret Thatcher on Europe was widely regarded as setting in train the process which led to her departure nine days later.
While those concerns are shared by Clarke, he said now was not the time for leadership election as he withdrew his call for a vote of confidence in the PM.
The border remains a crucial sticking point in Brexit negotiations with the European Union, amid a dispute on how to maintain free flow of movement across the 310-mile frontier between Northern Ireland and the Irish Republic. She also insisted that Britain would take a tough stance in future negotiations with the EU.
He apologised, claiming it was an "honest mistake" made by the Tory whips in "fast-moving circumstances", but Ms Swinson said it was deliberate "cheating".
Eurosceptic MPs had targeted her government's customs legislation to try to toughen up her plans to leave the European Union, but instead of facing them down and fuelling tensions, her spokesman said the government would accept their four amendments.
"The passing of this deeply flawed bill can't mask the profound splits at the heart of the Tory party", said Mr Brake.
Blair himself said it was a "mush", an "incomplete half-in half-out" plan that pleased no one - and was unlikely to be accepted by Brussels.
Shadow worldwide trade secretary Barry Gardiner said: "The government's handling of Brexit over the past week has been an utter shambles".
"We would look to be an active participant and this would involve making an appropriate financial contribution".