A plan by Brexiteer Tory MPs to force a vote of no-confidence lost -momentum yesterday as the number of letters submitted to Sir Graham Brady, chairman of the backbench 1922 Committee, failed to reach the 48 needed to trigger a ballot.
Friday brought some respite, as supportive Cabinet ministers rallied around her.
Tom Watson says it is time for Labour to be given a chance to rescue Brexit as he called for a General Election amid turmoil at the heart of Government.
In a public relations offensive, May revealed in a Daily Mail interview how her husband supported her during "a pretty heavy couple of days".
The plans include "buffer stocks so that we have all the logistical capacity that we need to carry on running our business", said Chief Executive Warren East.
Britain's Conservatives have been divided for decades over Britain's membership in the European Union, and the draft withdrawal agreement has infuriated the most strongly pro-Brexit members, who want the country to make a clean break with the bloc.
Gosport MP Caroline Dinenage says that although she hasn't seen the full deal yet, she will also be backing Theresa May in the coming months.
"We are determined as a Government to ensure that both British citizens living, working and travelling to Europe and European Union citizens coming to the United Kingdom are able to continue with the current healthcare arrangements".
"I'll be going through the detail, listening carefully to all the different views by people in Dudley, reading all the briefing and listening to the experts".
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Early in December, the prime minister will put her deal to a "meaningful vote", in parliament.
A Remain supporter during the 2016 referendum, Ms Rudd has given her backing to Mrs May's draft Brexit agreement, saying it is "not flawless but ideal was never on offer".
The Prime Minister's allies insist she can survive a no-confidence vote but it would leave her badly wounded.
Eurosceptic MPs fear the deal would keep Britain shackled to Brussels long after Brexit on March 29, 2019. One caller said May should resign and let a more staunchly pro-Brexit politician take over; another compared her to Neville Chamberlain, the 1930s prime minister who tried in vain to appease Nazi Germany to avoid war.
Mr McFadden said: "If she can not get the agreement through Parliament then I think we have to put the issue back to the people".
"If it comes to a confidence vote, she will have mine". Instead, Barclay - who supported Leave in 2016 - will focus on the domestic legislation to prepare Britain for its scheduled departure in March.
The resignations mean that the prime minister may face an uphill battle to win parliamentary support for the Brexit deal. She did not immediately address that part of the caller's question.
Ms Rudd, a key ally of Mrs May who stood in for her during last year's televised leaders' debate, was deployed within moments of her appointment to appeal for calm in the party.