Theresa May’s Northern Irish DUP partners in government have said they will not support her Brexit deal if it is put to a vote again in the House of Commons.

The announcement deals a devastating blow to Ms May’s hopes of getting her Brexit deal through the House of Commons at the third time of asking, with her government relying on DUP MPs for its majority.

It came just hours after Ms May had told Conservative MPs desperate to see the back of her that she would quit if it meant they backed her deal.

Any boost the prime minister received from her pledge was quickly neutralised when Arlene Foster told Sky News in an interview that her party will not be following Ms May through the voting lobby.

The DUP leader said: “What we can’t agree to is something that threatens the union, that has a strategic risk to the union.

“For us in the Democratic Unionist Party, the union will always come first and that has been the issue right from the beginning of all of this.”

The party’s leader in Westminster later underlined that the DUP would not simply abstain, but would vote against the deal again, making it incredibly difficult for it to pass even if Tory rebels change their minds.

In a statement the DUP said it had had good discussions with the government in recent days and some progress on domestic legislation had been made.

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It went on: “All concerned recognise the need to ensure that as we leave the European Union the economic and constitutional integrity of the United Kingdom is maintained.

“However, given the fact that the necessary changes we seek to the backstop have not been secured between the government and the European Union, and the remaining and ongoing strategic risk that Northern Ireland would be trapped in backstop arrangements at the end of the implementation period, we will not be supporting the government if they table a fresh meaningful vote.”

They argued that the backstop – an arrangement that comes into play if the UK and EU have failed to agree a deal by December 2020 – has the potential to create an internal trade border down the Irish sea.

The party added: “We want to secure the United Kingdom’s departure from, and our future relationship with, the European Union on terms that accord with our key objectives to ensure the integrity of the United Kingdom.