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The main contenders for the overall title crossed the line together nearly three minutes after Bilbao. You have to go back to 2018 and Omar Fraile's win for the last victory by a Spanish rider.
Bilbao dedicated the victory to Gino Maeder, his Bahrain Victorius team-mate who died last month after a crash on the Tour of Switzerland.
"I had to do it for Gino," said Bilbao.
Maeder, a 26-year-old Swiss rider, plunged into a ravine while competing in his home race in June, and succumbed to his injuries a day later.
"He was such a great guy and we are trying to honour his legacy with charity work - this is our project to remember him, and not just by sports," said Bilbao.
He was delighted with his first Tour stage win.
"It's an incredible feeling, I have waited years for this," he said. "It would have been a dream to win at home because we started in Bilbao.
"I was lucky this stage came after a rest day, but the day off meant I was fully hydrated."
The Spaniard led a chase to track down Latvia's Krists Neilands, who looked set to make it two stages in a row for Israel-Premier Tech before he was finally caught by a six-man pursuit.
- Sizzling conditions -
On a baking 167km run from the volcano-themed Vulcania amusement park billed as one of the Tour's most beautiful stages, the peloton laboured over five hills to the village of Issoire in Auvergne.
The pack splintered due to the intense conditions in exposed terrain on the semi-deserted hillsides along the Puy de Dome tectonic fault line.
Defending champion Jonas Vingegaard retained his 17-second lead in the overall standings but second-placed Tadej Pogacar forced the Dane to chase him early in the stage.
Vingegaard said he was happy for the Bahrain team.
"I feel for them today and for the family of Gino. I'm happy to see they are doing well, I guess they are doing it for Gino," he said.
Pogacar himself looked relieved the stage was over.
"The team did a great job keeping me cool," said the Slovenian, who unlike Vingegaard, dislikes extreme heat.
"After that the rest day seems like a long time ago," Pogacar said.
American Neilson Powless also held on to the polka dot jersey for the most climb points and will likely do so again Wednesday.
"We need that day off. I didn't do anything, and you should have seen how much water was being taken on today," said Powless.
Riders took on between 12 and 15 water bidons each to stay hydrated during the searing stage.
Jasper Philipsen retains the green jersey ahead of stage 11, which offers the Belgian Alpecin team's fast man a chance at a fourth stage win.
"I was putting ice in my vest all day, it was a question of hydrating but they were trying their best to keep us cool too," said Philipsen.
"We nearly caught the escape, we got the gap down to two minutes but tomorrow's another day and the goal is a fourth win."