Boris Johnson’s new government could struggle to survive after 17 senior ministers were sacked or resigned from government, leading parliamentary sources have warned.

A “Gaukward squad” of four former cabinet ministers, led by the ex-chancellor Philip Hammond, have indicated that they are determined to block a no-deal Brexit after resigning from government on Wednesday before they could be sacked.

In a major purge of Theresa May’s ministers, Johnson sacked 13 other prominent ministers, including Brexiters, who had backed his leadership rival Jeremy Hunt.

Nigel Evans, a member of the backbench 1922 Committee, claimed it was “not so much a reshuffle as a summer’s day massacre”.

Tom Watson, Labour’s deputy leader, wrote on Twitter that Johnson’s decision to purge so many ministers could topple his administration. “This huge cull will lead to early collapse of Johnson’s govt bringing GE [general election] that much closer,” he tweeted.

Hunt, the foreign secretary, decided to walk away from government after being offered the role of defence secretary.

Johnson’s government will have a majority of just two if, as expected, the party loses the Brecon and Radnorshire byelection in August to the Liberal Democrats. His majority was eroded further on Monday by the withdrawal of the whip from Charlie Elphicke, a Tory backbencher charged with three counts of sexual assault against two women.

The former chancellor Hammond is expected to lead resistance to a no-deal Brexit among senior former ministers. Others include David Gauke, the former justice secretary, Rory Stewart, the former international development secretary, and David Lidington, who left the Cabinet Office.

All are resolute opponents of a no-deal Brexit, and would join a group of former ministers including Sir Alan Duncan, Steve Brine, Anne Milton and Richard Harrington who have resigned in recent months because of their opposition to no deal.

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It remains unclear whether May will join them after stepping down from office. She is believed to be concerned about the future of the union of the UK if Johnson leaves the EU without a deal, but pledged to support the new PM in a speech to parliament.

Johnson has repeatedly promised to take the UK out of the EU on 31 October, with or without a deal. Speaking outside No 10, the new Tory leader said he would meet the deadline, “no ifs or buts”.

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Johnson and his new team will also have to win over other ministers who have been sacked from the cabinet. Others who could prove difficult opponents include the former defence minister Penny Mordaunt, the former international trade minister Liam Fox, and James Brokenshire, who departed as communities secretary despite backing Johnson’s leadership campaign.

Others forced out on Wednesday include Greg Clark, the business secretary, Karen Bradley, the Northern Ireland secretary, the education secretary, Damian Hinds, and the Scottish secretary, David Mundell.

Hammond said on his resignation that the new prime minister should be “free to choose a chancellor who is fully aligned with his policy position”. In a pointed message to Johnson, he said the headroom built up in the public finances could only be used for tax cuts and spending increases if a Brexit deal were secured.

“After a decade when the aftermath of the 2008-09 recession meant we had no choices, this is a luxury which our successors should use wisely,” he said. Hammond handed his resignation to May after her final prime minister’s questions.

It is highly unlikely that MPs could move against no deal before they break up for their summer recess at the end of July. A move is more likely during a narrow window of opportunity in September or October. Tory MPs including Dominic Grieve and Oliver Letwin are liaising with colleagues and Labour frontbenchers over the best tactics.

Labour concede that it is unlikely that a motion of no confidence could be tabled before the summer recess.

The Tory rebels are prepared to give Johnson the summer to see if he can make headway towards coming to a fresh agreement with the EU that avoids no deal, but will make their moves against him after that if he takes the government full tilt towards a no-deal Brexit.

One former minister said: “More than 40 Conservatives voted against no deal under May. Most are prepared to put everything on the line to stop a Boris no-deal.”

The cabinet departures followed other key resignations including Mick Davis, the chief executive and treasurer of the Conservative party, who also stood down on Wednesday morning. “One of my greatest bugbears has been the lack of unity in our party,” the mining executive – one of the party’s most generous donors over the past six years – wrote to Conservative benefactors on Wednesday.

Johnson is expected to appoint another chief executive within days. John Griffin, the minicab tycoon who has given the party more than £4m over the past six years, said he would consider taking up the role if it were offered.