Theresa May dropped the clearest hint yet that Britain will still pay into the EU budget after Brexit – despite being told it would be “betraying” voters.

Delivering a statement on last week’s EU summit, the Prime Minister still gave little away about her Brexit strategy, refusing to tell MPs when it would be published.

And she batted away questions about single market membership as inquiries about “means” when what mattered was her end game of securing “the best possible deal”.

However, Ms May did appear to confirm that she is willing to continue paying into the EU budget, possibly as part of a transitional deal to cushion the exit.

The move will be controversial both with hardline Brexit supporters – and with voters who were told that £350m a week more will be given to the NHS instead.

Ms May was put under pressure by Conservative backbencher Philip Davies, who said funding for the EU was “one of the big issues” in the referendum campaign.

He said: “Will the Prime Minister make a pledge that, when we leave the EU, we will not paying any money into the EU budget?

“Surely even contemplating that will be contemplating betraying what people voted for?”

But Ms May refused to give such a pledge, saying instead: “What’s important is that, when we leave the European Union, people want to ensure that it’s the British Government that decides how taxpayers’ money is spent.”

Other Cabinet ministers, led by Chancellor Philip Hammond, are calling for a transitional Brexit deal – which would require payments – but Ms May has refused to back her key colleague publicly.

In Brussels on Thursday, the other EU leaders spent just 20 minutes discussing Brexit – after Ms May had left for her RAF plane back to Britain.

They refused Ms May’s request for a quick decision on expat rights after Brexit until after Article 50 has been invoked, early next year.

And the Prime Minister was warned that a £50bn “Brexit bill” – for outstanding liabilities – would be “one of the first issues on the table” in the negotiations.

In the Commons, a string of right-wing Tory backbenchers urged Ms May not to give ground in the talks and to move ahead as quickly as possible.

Peter Lilley, the former Conservative Cabinet minister, demanded a speedy departure claiming every week Britain remains in the EU costs the country £250m.

Ms May again vowed to trigger Article 50 by the end of March – even if the Supreme Court rules that Parliament must give its approval.

And she hinted she wants Britain to remain in Europol and keep the European arrest warrant, after Brexit – having argued for them as Home Secretary.

But her refusal to give any further details of her economic Brexit strategy was strongly criticised by Pat McFadden, a pro-EU Labour MP.

He said: “The Prime Minister’s New Year’s resolution must be to give more information to Parliament and the people about what the Government actually wants from Brexit.