Jeremy Corbyn has been given a boost in his bid to prove his ability to win over Tory voters after Labour won a council seat off the Conservatives in a by-election.

It was the first victory at the ballot box since Mr Corbyn won a landslide victory in the Labour leadership election last month. The party won a marginal ward on the Conservative-controlled Cherwell District Council on a swing of more than 6 per cent. Voting in the Banbury Grimsbury & Castle seat in north Oxfordshire followed the death of a Conservative councillor.

In the three other local by-elections since Mr Corbyn was elected Labour leader the party has held the seats comfortably, but the Cherwell vote will be used by Corbyn supporters as proof he can win over crucial Tory votes needed to win in 2020.

Moderates in the Labour party have warned that shifting further left under Mr Corbyn will leave the party with even less chance of winning back centrist voters who gave the Tories a slim majority at the general election after being put off by Ed Miliband’s interventionist and anti-business policies.

Several leading figures, including Tristram Hunt and Dan Jarvis, said at the Labour party conference in Brighton earlier this week that next May’s London Mayoral elections and the elections in Scotland and Wales will deliver a crucial verdict on Mr Corbyn’s ability to win in 2020 and hinted that failure to perform well could spell the beginning of the end for Mr Corbyn’s leadership.

Lord Mandelson has warned moderates not to try to oust him too early, however, as doing so could easily lead to Mr Corbyn being replaced by a similarly left-wing figure. In a leaked paper circulated to Blairite allies, he advised Labour MPs to let Mr Corbyn prove his “unelectability” before making a move.