Gina Miller, the pro-Europe campaigner, has launched a tactical voting drive in Theresa May’s own seat amid claims that tightening polls show there is no mandate for a “hard Brexit”.

Voters in the prime minister’s Maidenhead constituency, which backed Remain in the EU referendum, will be targeted with material urging them to vote for May’s Lib Dem rival to “send a message” that there is no appetite for the terms of Brexit she has set out.

The move comes as polling for Miller’s Best for Britain group suggests most voters would accept a deal that handed EU citizens the right “to travel, work, study or retire” in Britain in exchange for EU countries giving Britons the same rights. The YouGov poll found that 62% said the government should pursue such a deal, with just 17% against. It found that 50% of voters said the government should aim for a deal to stay in the EU’s single market, with 21% against the idea.

“If May ploughs on irrespective of the narrowing of the polls and our research, then I think the public has every right to say that she is going against the will of the people, and this is her personal version of Brexit,” Miller told the Observer. “Is she trying to appease Ukip voters and the right of her party, or she is she a Ukipper in disguise?”

Attempts to form progressive alliances and encourage tactical voting to limit the Conservative majority have been floundering, largely due to Labour’s refusal to cooperate with local pacts, and the Tories’ huge poll lead early on.

However, the latest polls and May’s decision to put Brexit back at the forefront of her campaign have breathed new life into such efforts, with several groups promising a last-minute increase in online targeting of voters.

Miller’s campaign, which is armed with more than £400,000 in donations, is now backing 36 candidates, including Tony Hill, the Lib Dem candidate in May’s constituency. The Lib Dems finished a distant third in Maidenhead in 2015, a full 30,000 votes behind May. However, Miller said an increased vote for the Lib Dems would send “a clear message that there is a percentage of constituents who are going to make their voices heard”.

She said Maidenhead was the most regularly searched constituency in her group’s online tool designed to help people decide how to use their vote in a tactical way. “The polling we’re getting is that Maidenhead is not supporting a hard Brexit. Mrs May, representing them in parliament, is pursuing that. There are lots of places where people voted Remain, or did not back an extreme Brexit, where the representative thinks they can do that and ignore the people. People are now asking questions. It is not this hard Brexit blank cheque that May wants.

“She has kept saying she wants a mandate, and it is a mandate for extreme Brexit and it is the will of the people. Our polling shows there isn’t this mandate for extreme Brexit and there is no mandate for their manifesto.”

Theresa May listens to members of the public during a general election campaign tour stop at Pot House Hamlet in Silkstone, on 3 June. Photograph: Matthew Lloyd/EPA

Help will also be given to Vince Cable, who is said to be struggling to regain the Twickenham seat he lost in a shock result in 2015, and Dr Louise Irvine, the GP attempting to defeat health secretary Jeremy Hunt in South West Surrey.

Miller’s campaign has gathered pace as evidence emerges that a fifth of all voters are planning to back someone other than their first-choice candidate – a huge increase on the 9% who indicated they would vote tactically in 2015.

The research by the Electoral Reform Society is not necessarily good news for those opposing Brexit. Ukip voters were the most likely to deploy a tactical vote, suggesting many will lend support to May as a result of her pledge to deliver Brexit and end freedom of movement.

Other tactical voting groups are also making a final push. The Progressive Alliance campaign, which is backing its own slate of candidates, is targeting 31 constituencies where it believes tactical voting can keep out the Conservative MP. Its campaign videos have been viewed 1.4 million times, and 100,000 people have used its Vote Smart tool. Frances Foley, from the campaign, said an operation in the Norwich South seat held by Labour’s Clive Lewis had been asking voters displaying Green and Lib Dem material to back Labour. “What we’re seeing is a ramping up of efforts in the final days – canvass sessions, door-knocking, street stalls.”

Banksy also provided a boost to tactical voting campaigners on Saturday by offering a limited edition artwork to voters in certain constituencies that voted against the Conservatives.

“Simply send in a photo of your ballot paper from polling day showing you voted against the Conservative incumbent and this complimentary gift will be mailed to you,” he wrote on his website. “Lawyer’s note: this print is a souvenir piece of campaign material, it is in no way meant to influence the choices of the electorate, has no monetary value, is for amusement purposes only and is strictly not for resale. Terms and conditions to follow, postage not included.”

The Lib Dem leader Tim Farron has also issued a last-minute plea for people to vote tactically. “There are dozens of seats across the country where the only candidate that can beat the Conservative is a Liberal Democrat,” he said. “If you live in one of those places, I need you to lend me your vote.”

GINA MILLER’S VOTING GUIDE FOR A SOFT BREXIT

Brentford and Isleworth

Ruth Cadbury (Labour)

Bridgend

Madeleine Moon (Labour)

Brighton Kempton

Lloyd Russell-Moyle (Labour)

Brighton Pavilion

Caroline Lucas (Green)

Bristol East

Kerry McCarthy (Labour)

Cardiff Central

Jo Stevens (Labour)

Cardiff West

Kevin Brennan (Labour)

Carshalton and Wallington

Tom Brake (Lib Dem)

Ceredigion

Mark Williams (Lib Dem)

Cheadle

Mark Hunter (Lib Dem)

Cheltenham

Martin Horwood (Lib Dem)

Ealing Central and Acton

Rupa Huq (Labour)

East Devon

Claire Wright (Independent)

Hammersmith

Andy Slaughter (Labour)

Hampstead and Kilburn

Tulip Siddiq (Labour)

Harrow West

Gareth Thomas (Labour)

Hazel Grove

Lisa Smart (Lib Dem)

Hove

Peter Kyle (Labour)

Isle of Wight

Vix Lowthion (Green)

Kensington

Emma Dent Coad (Labour)

Kingston & Surbiton

Ed Davey (Lib Dem)

Leeds North East

Fabian Hamilton (Labour)

Lewes

Kelly-Marie Blundell (Lib Dem)

Maidenhead

Tony Hill (Lib Dem)

Norwich South

Clive Lewis (Labour)

Portsmouth South

Gerald Vernon-Jackson (Lib Dem)

Richmond Park

Sarah Olney (Lib Dem)

St Albans

Daisy Cooper (Lib Dem)

Sheffield Hallam

Nick Clegg (Lib Dem)

South West Surrey

Dr Louise Irvine (NHA)

Sutton and Cheam

Amna Ahmad (Lib Dem)

Vauxhall

George Turner (Lib Dem)

Streatham

Chuka Umunna (Labour)

Tottenham

David Lammy (Labour)

Twickenham

Vince Cable (Lib Dem)

Westminster North

Karen Buck (Labour)