Sunday is the first day of the Conservative Party Conference in Birmingham. But with just six months left on the Brexit clock – and the day’s papers full of Tory divisions on Theresa May’s Chequers deal – the Sunday politics programmes were once again dominated by the subject of leaving the EU.

Leading the way was the Prime Minister herself, who – during an appearance on the BBC’s Andrew Marr Show – refused to say whether she would compromise on the Chequers proposal for Brexit in order to avoid no deal.

Repeatedly telling Marr she wants to hear the EU’s “detailed concerns” on the proposed deal following a savage brush-off by leaders in Salzburg earlier this month, May said: “It’s an ambitious arrangement.

“The reason the European Union has been saying that they have concerns with it is precisely because this is an ambitious arrangement for the United Kingdom for the future.”

The PM also refused to be pinned down on the issue of Northern Ireland and how the government would be able to prevent a hard border in the case of no deal. 

Insisting that the party is “very clear” it does not want a hard border between Ireland and Northern Ireland, May said the Conservatives would do “everything we can” to prevent one in the case of no deal. 

But the Tory leader seemed unable to explain how they could ensure this, ignoring Marr’s arguments that World Trade Organisation rules dictate that such a border needs to exist if the UK crashes out of the EU. 

“We will make a success of Brexit, regardless of the outcome of negotiations,” May added.

The interview ended on a frosty note after Marr quizzed the PM about her role in the Windrush scandal, asking how the hostile environment policy she oversaw as Home Secretary had affected the Windrush Generation. 

With May unwilling to accept any real responsibility for the scandal, the episode ended with the uncomfortable-looking pair sat in silence as the credits rolled. 

But despite claims from Labour on Saturday that the Conservative Party is the “most divided government ever”, the Prime Minister received backing from Liam Fox on her Brexit plans when he spoke to the Sunday press. 

The international trade secretary told BBC 5 Live’s John Pienaar it’s “important to support the government’s position” on Brexit. 

“And the reason that it’s important to do that is that we now need to put pressure on the European Union to come up with a set of proposals,” Fox said. 

“Taunting the Prime Minister as they did post-Salzburg… is not an appropriate way to proceed.” 

Scottish Tory leader Ruth Davidson also offered May her support during an interview with Sky’s Sophy Ridge, arguing that the “blow-up” in Salzburg had actually “cleared the air” between the EU and the UK.

Calling for the party to give the PM space to “get in and get the job done”, Davidson said: “I think some of the noisers-off perhaps need to calm down.”

Her comments come after former foreign secretary Boris Johnson appeared to accuse May – who campaigned for remain in the 2016 referendum – of not fully supporting Brexit, calling her plans “deranged” in the Sunday Times.

Davidson criticised Johnson’s attempt to distance himself from the Chequers proposal, adding: “I know what was being said in December – I’m not sure how the former foreign secretary didn’t.” 

Former Brexit secretary David Davis – who dramatically resigned this summer to protest against the government’s stance on Brexit – was less positive about May’s strategy when also appearing on Ridge’s show.

Asked how he would describe May’s Chequers plan, he replied: “Just wrong”.

“If we got Chequers, the Europe issue wouldn’t go away,” Davis said. “The toothache we’ve had for 30 years on this subject would still be there for another 20 years.”

But while Davis worked as Brexit secretary for two years under May, he refused to accept responsibility for the government’s proposal, admitting only that he “failed to persuade the Prime Minister”.

However, the Haltemprice and Howden MP insisted there is still “plenty of time” to save the Brexit deal, telling Ridge there is an “80% to 90%” chance the UK will end up with a free-trade-based deal.

“But we’re going to have a very scary few weeks,” Davis continued. “Between now and about November, it’s going to be really scary. Everybody is going to be calling each other’s bluff, all sorts of brinkmanship is going to be going on.

“That’s normal, that’s the European Union’s daily bread,” he said, adding that the EU likes to pile on time pressure when reaching a deal. “That’s what we’ve got to be ready for.”

But while Davis was quick to disparage May’s Brexit strategy, he said he would back her in a conference vote on her leadership, saying: “I want a change of policy, not a change of leader”.

Questioned about whether he believes Johnson would be a good leader, he told Ridge: “That will probably be after I’ve gone”.

“Boris is a great mate of mine,” Davis added. “But a lot of his ideas are good headlines, not necessarily good policies.”

His comments come after an exclusive HuffPost UK poll revealed that the former London mayor is a firm favourite to replace May if she was to step down – but is less popular than Jeremy Corbyn and would probably lose a general election to Labour.

While most of his Tory colleagues were questioned over Brexit, Conservative chairman Brandon Lewis was quizzed by Ridge over the other big story of the day – a huge security flaw with the Tory conference app which allowed users to access the phone numbers of MPs.

When asked whether he had considered resigning over the controversy, Lewis said that “at the moment” he was focusing on making sure conference goes well.

“Any breach of data is a serious matter,” he told Ridge, before adding: “Within half an hour, we had fixed this problem.”