Primoz Roglic extended his lead in the Tour de France on Friday as Daniel Martinez scored a first stage win for US team Education First by triumphing in a titanic mountain battle.

Defending champion Egan Bernal dropped a further 37 seconds on his two key rivals, the Slovenian pair of Roglic and Tadej Pogacar, over the final kilometer of the 191.5-kilometer 13th stage raced across the Cantal volcano chain.

Martinez came under extreme pressure for the stage win from fellow breakaway riders Maximilian Schachmann and Lennard Kamna as the Bora pair broke for the winning line at 1,589 metres of altitude.

Gritting his teeth and pouring with sweat, the Colombian however dug deep for a final kick that carried him to a last-gasp win.

"I managed to hold on and win with a late sprint," said Martinez, who formed a heart with his fingers on the finish line.

Martinez won the key Tour de France warm-up, the Criterium du Dauphine, in August and also a stage on Paris-Nice.

"I did well on those races, but winning a stage on the Tour is on another level," said the 24-year-old.

Further down the hill as the race hit a nine-percent incline on the day's sixth and final climb, Roglic and Pogacar managed to shake off an elite clique of riders in contention to win this Tour.

Slovenia vs. Colombia

Roglic now enjoys a 44sec lead over 21-year-old Pogacar, who climbed to second, while Bernal dropped to third, 59sec off the yellow jersey.

"It was a Slovenian day, there's two of us ahead of a few Colombians," Roglic said.

"There are great professionals in Slovenia in all kinds of sports, it's a tough nation and we are all pushing each other.

"Looking at it from distance this Tour is not finished and (Bernal) is not finished, but I don't want to bother myself worrying about other people," Roglic said.

"When I saw that final climb I thought 'pfffft, this is going to hurt'. So I'm happy it all went my way."

Defending champion Bernal now looks under severe pressure from Roglic, who has a powerful climb team around him.

The director of Bernal's Ineos team, Benjamin Rasch, said the 23-year-old was improving each day from a bad back and that Friday's stage did not suit his leader.

"We will do better on the really tough stages," Rasch said.

"Of course he wants to win the tour," Rasch said "He's a good leader and a good lad."

Pogacar, two years younger than Bernal, also reclaimed the best under-25s white jersey thanks to his Puy Mary climb.

"I gained time on everyone except Roglic on the last hill and yes we are both Slovenians but we were not friends over those final kilometres," said Pogacar, of the UAE team.

Concussion forces Bardet out

The Puy Mary proved to be something of a Tour de France femme fatale for two of the host nation's top hopes.

Romain Bardet started the day in fourth just 30 seconds off the lead in a region where his grandmother lives, but his luck ran out with a heavy fall that subsequently forced him to quit this year's Tour with concussion.

Bardet, who was King of the Mountains last year and has twice finished on the podium at the Tour, ended Friday's stage 11th overall and three minutes off the pace after crashing alongside Bauke Mollema, who also retired but with a fractured wrist.

"The fall was violent, downhill, and I struggled all day," said Bardet, who was France's best hope of an overall winner after Thibaut Pinot's dramatic stage eight meltdown.

"The medical examinations confirmed what I was expecting and I am unable to continue the race."

The other remaining home hope, Guillaume Martin, faded on the penultimate climb after starting the day third, he ended the day just below Bardet at 3min 14sec.

With no cars allowed on the Puy Mary climb thousands of fans hiked to the summit to greet the riders on another day of blockbuster crowds.

The holder of the green jersey, Irishman Sam Bennett, and the vastly experienced Peter Sagan will likely renew their struggle for sprint points on Saturday's 194-kilometre run over five small hills between Clermont-Ferrand and Lyon.

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Tour de France: Primoz Roglic tightens grip as defending champion Egan Bernal falters