On two of Monday's votes her majority was cut to three, suggesting that the leader will struggle to get Brexit legislation through a deeply divided parliament, which could possibly threaten the approval of any Brexit deal with the EU.
The government said being part of an EU customs union would prevent it from striking global trade deals, won the vote by 307 to 301, a slim majority for May of just 6.
International Trade Secretary Liam Fox told parliament the government's future trade bill was "an important bill providing continuity and stability".
The Trade Bill is focused on converting trade deals between the European Union and third countries into bilateral deals with Britain. "Yesterday changed that", MP Phillip Lee, who resigned as a junior minister over Brexit earlier this year, told lawmakers.
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Sir Bernard told the programme that the Chequers plan was "dead", with opposition from both wings of the Tory party. The new UK Brexit secretary Dominic Raab will meet the EU'S chief negotiator Michel Barnier on Thursday (19 July).
This was underlined on Monday as it first accepted a series of demands from Brexiteers who are unhappy at the proposals in the White Paper, believing it keeps the United Kingdom too closely tied to the EU.
In a sometimes heated debate Conservative Anna Soubry said the plan May agreed at her meeting with ministers at her Cheqeurs retreat had now been wrecked by caving in to no-deal Brexiteers. "If we don't, we risk ending up with no Brexit at all", she wrote in an article in the Mail on Sunday newspaper.
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.
Facing defeat because of opposition from dozens of pro-Brexit MPs, the government agreed to changes, including a clause to toughen conditions over customs arrangements and another to ensure the United Kingdom is out of the EU's Value-Added Tax system.
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France had mobilised around 110,000 security staff for the Bastille Day festivities on Saturday and the World Cup final. "We mix. France's World Cup win saw celebrations in the country descend into violence as riot police used tear gas to control crowds.
May had to fight hard to get the agreement of cabinet ministers at her Chequers country residence for her vision for Britain's future ties with the European Union, only for it to be undermined by the resignations last week of her Brexit minister David Davis and foreign secretary Boris Johnson.
But this angered MPs from the party's pro-EU wing who refused to back the new amendments, leading to heated exchanges in the House of Commons as the Customs Bill was debated.
The government has so far provided no reason for the motion, which Labour is reportedly set to vote against.
Labor's Brexit spokesman Keir Starmer said: "It was a very significant defeat for the government tonight on European medicines regulation amendment and a near miss on customs union amendment".
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Kasparov said that the press conference was a clear indication Trump and Putin are alike in that neither represent their nations. Monday's news conference turned on some sharp questions by Jonathan Lemire, White House correspondent for The Associated Press.
The votes were preceded by frantic activity in the Commons, with live negotiations conducted on the floor of the chamber as Government whips led by Julian Smith made approaches to rebel MPs. Critics say the move is a bid by a government to escape the prospect of more rebellions in the Conservative Party.