At Ron DeSantis’s election victory party in Tampa on Tuesday night, supporters of the rightwing Florida governor chanted: “Two more years!”

Governors serve four-year terms, but DeSantis is widely seen as a possible challenger to Donald Trump for the Republican presidential nomination in 2024. DeSantis’s strong performance in Florida on Tuesday – as other Republicans across the US faltered – has greatly strengthened that position.

In a landslide victory, with more than 95% of votes in, DeSantis won 59.4% of the vote with 4,607,597. Meanwhile, the Democratic candidate, Charlie Crist, won just 40% of the vote with 3,100,603.

Trump has been reported to be planning a 2024 announcement this month, seeking to capitalise on Republican success in the midterm elections. At a rally in Ohio on Monday, he trailed an announcement on 15 November.

But on the night when the Republicans’ hoped-for “red wave” seemed unlikely – though control of the House and the Senate remained in the balance – the atmosphere at Trump’s Florida home, Mar-a-Lago, was reported to be anxious.

In Tampa, meanwhile, celebrations of DeSantis’s convincing win over his Democratic challenger Crist were raucous.

In a speech laden with allusions to his handling of the Covid pandemic – a source of great controversy and under which 82,541 Floridians have died, the third-highest total of any US state – DeSantis said: “We chose facts over fear, we chose education over indoctrination, we chose law in order over rioting and disorder.

“Florida was a refuge of sanity when the world went mad. We stood as the citadel of freedom for people across this country and indeed, across the world. We faced attacks, we took the hits, we weathered the storms but we stood our ground.

“We did not back down. We had the conviction to guide us and we had the courage to lead. We made promises. We made promises to the people of Florida and we have delivered on those promises. And so today, after four years, the people have delivered their verdict.”

DeSantis has refused to say if he intends to serve a full second term, awkwardly so in his debate with Crist. In Tampa, when supporters chanted “two more years”, the governor smiled broadly and said: “Thank you very much.”

He did not address Trump at all, let alone the former president’s threat, reported earlier in the day, to reveal “things about him that won’t be very flattering” if DeSantis does mount a White House run.

Polling of the notional Republican field for 2024 gives Trump big leads but DeSantis is the only other name to regularly attract double-figure support. The Florida governor regularly wins polls when Trump is left out.

News outlets have reported that DeSantis has indicated to donors he may seek to avoid confronting Trump, waiting for 2028 instead.

But the measure of DeSantis’s success on Tuesday, coupled with an easy win for the Republican senator Marco Rubio over his Democratic opponent, the congresswoman and former Orlando police chief Val Demings, may have changed the equation.

In Tampa, DeSantis nodded to startling successes in previously solidly Democratic areas, most prominently Miami-Dade county, when he said: “Thanks to the overwhelming support of the people of Florida we not only won the election, we have rewritten the political map. Thank you for honoring us with a win for the ages.”

He also nodded to policies that have proved controversial but profitable, particularly regarding Covid and cracking down on the teaching of LGBTQ+ issues and the history of race in America.

“We have embraced freedom,” he said. “We have maintained law and order. We have protected the rights of parents. We have respected our taxpayers and we reject ‘woke’ ideology.”

The speech had the ring of a politician with one eye on the national and therefore global stage. Given conservative Americans’ long-established veneration of Winston Churchill, it was perhaps not surprising that in a speech delivered with notable confidence, DeSantis echoed the wartime British prime minister.

In a speech largely devoted to crowing over the American left, DeSantis echoed Churchill’s famous promise from 1940, when he said Britain would fight “on the beaches … on the landing grounds … in the fields and in the streets”.

“We fight the woke in the legislature,” DeSantis said, to steadily mounting cheers. “We fight the woke in the schools, we fight the woke in the corporations. We will never ever surrender to the woke mob. Florida is where woke goes to die.”