The unity of Theresa May’s cabinet has publicly crumbled after eight of her most senior ministers refused to back her plan to delay Brexit by three months.

Those who failed to support it included the prime minister’s Brexit secretary Stephen Barclay, who was joined by 187 other Conservative MPs and frontbenchers in voting against her approach.

None of the ministers opposing Ms May’s ultimately successful move to delay Brexit will be sacked, because she allowed a “free vote” on the issue in the face of a mass rebellion.

It followed a rowdy cabinet meeting in the morning where tempers frayed over a separate rebellion the night before in which remainer cabinet ministers refused to vote with the government on a motion ruling out a no-deal Brexit.

As a result of Thursday night’s votes, however, the path is cleared for Ms May to put her twice-rejected Brexit deal to a third vote of MPs, and if it is passed she will request a Brexit delay until 30 June to push through necessary legislation.

Discussions between Conservative ministers and the party’s Northern Irish DUP allies in government continued as Ms May hunts for a majority for her agreement.

After losing on previous occasions in the Commons, the prime minister was forced to let MPs vote on extending Article 50 on Thursday.

The motion put in front of the house said she would go to the European Council at the end of next week and ask for a short delay to the end of June if MPs backed her deal, and a longer one if they did not.

But with Conservative ranks full of MPs who baulk at the idea of delaying the UK’s departure at all, and the prospect of having to sack dozens of minsters, she allowed a free vote.

As well as Mr Barclay, top ministers who voted against her approach included defence secretary Gavin Williamson, trade secretary Liam Fox, commons leader Andrea Leadsom, Treasury secretary Liz Truss, transport secretary Chris Grayling and development secretary Penny Mordaunt. Welsh secretary Alun Cairns abstained.

A further 27 junior ministers and 11 whips also voted against the approach Ms May is taking, including the outspoken Conservative deputy chairman James Cleverly and other Brexit ministers Kwasi Kwarteng and Chris Heaton-Harris. Three other whips, including chief Julian Smith, declined to vote.

A Downing Street spokesman sought to play down the splits in the cabinet and wider party, pointing out that the prime minister had granted them a free vote so it was normal that they could chose to act they wished.

He went on to point the finger of blame squarely at the Commons, when discussing why the delay was necessary: “The prime minister absolutely wanted and strived for the United Kingdom to be leaving the EU on 29 March.

“Everything she has done since she came to office was intended to deliver that. She did not want there to be an extension. She brought forward a meaningful vote twice having agreed a deal with the EU in November.

“Parliament chose to reject that deal. So we now have to confront the difficult position that decisions taken by parliament have left us in.”

Earlier in the day, tensions peaked when Ms May sat at the cabinet table to discuss events the night before, when remainer cabinet ministers Amber Rudd, David Gauke, Greg Clark and David Mundell rebelled against her orders that they should oppose a plan to strongly rule out a no-deal Brexit.

Reports suggested that Mr Smith attacked the remainers, and then stormed out when business secretary Mr Clark tried to explain himself.

Following the bust up, Ms May agreed to allow a free vote on Thursday’s motion on delaying Brexit in an apparent attempt to keep Brexiteer ministers happy and avoid a damaging split.

In other votes on Thursday, the prime minister’s Conservative MPs did follow orders allowing her to successfully block opposition proposals that would have allowed parliament to seize control of Brexit and decide what to do next.

The result confirms that Ms May will now try to win a third vote on her Brexit deal, after it was defeated by 149 votes on Tuesday and 230 votes earlier in the year.

The Independent understands that Ms May will likely schedule the vote for Tuesday next week, allowing her a few days to try to convince the DUP and Eurosceptic Tories to back it.

Discussion with the DUP were set to continue through the weekend, with the party’s leader Arlene Foster said to have not yet bought her ticket back to Belfast.

She said: “We are talking to the government and to the attorney general at the moment to try and make a deal happen because we want to see Brexit working, we want to see it working for the whole of the UK in a way that doesn’t leave Northern Ireland behind.”

The prospect of a lengthy delay to Brexit had also prompted some Tories to admit that they might now back Ms May’s deal.

Conor Burns, a close ally of Boris Johnson, said: “I am now looking for a reason to support it. The pretence that this parliament has honoured this referendum result is now slipping away.

“If there is a two-year extension then your [Brexit] mandate is five years old and the urgency of your instruction clearly diminishes over time.”

Another, Lucy Allan, wrote on Twitter: “Last night, Labour MPs together with a handful of hardcore Conservatives voted to take no deal off the table. So now we will not be leaving the EU on 29 March…and they cheered.

“Now the option we are faced with is Theresa May’s deal or no Brexit.”