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Switzerland stunned France early in the first half on Monday after Haris Seferovic fired his side into the lead at the National Arena in Bucharest, Romania.
Karim Benzema's superb quick-fire brace in the second half pulled the 2018 World Cup champions back into the contest.
France, who were one of the favourites to win Euro 2020, took control of the last-16 clash in the 75th minute thanks to Paul Pogba's sensational strike from outside of the box.
However, Switzerland rallied back and Benfica striker Seferovic pulled one back for the Swiss in the 81st minute.
Mario Gavranovic was played through and kept his composure as he expertly finished past France captain Hugo Lloris ahead of the full-time whistle.
Neither France nor Switzerland could break the 3-3 scoreline in extra time, which forced the thrilling tie to head to a penalty shootout.
Pogba, Olivier Giroud, Marcus Thuram and Presnel Kimpembe all converted from the spot-kick
However, Mbappe stepped up for his penalty and could only watch on as Yann Sommer saved his effort, with Switzerland knocking France out of the round of 16.
The 22-year-old Paris Saint-Germain superstar has now taken to Instagram and admitted he "failed" his team.
Mbappe insisted that the sadness from France's exit is "immense," but he vowed the French would bounce back "even stronger for the next challenges."
The France international wrote to his 53m followers: "Very difficult to turn the page. The sadness is immense after this elimination, we could not reach our goal.
"I'm sorry for this penalty miss. I wanted to help the team but failed. Getting to sleep will be difficult, but unfortunately it is the ups and downs of this sport that I love so much.
"I know you fans have been disappointed, but I would still like to thank you for your support and for always believing in us.
"The most important thing will be to get up even stronger for the next challenges. Congratulations and good luck to Switzerland."
Switzerland will now face Spain in the quarter-finals on 2nd July at the Gazprom Arena in Saint Petersburg, Russia.
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