His Highness Sheikh Hamdan bin Mohammed bin Rashid Al Maktoum, Crown Prince of Dubai and Chairman of Dubai Executive Council, posted a picture with Reffet saying "We will miss you".
In a statement posted on social media, the company said the 36-year-old French stuntman Vincent Reffet died "during training in Dubai" on Tuesday.
It added that he was a "talented athlete, and a much-loved and respected member of our team...Our thoughts and prayers are with his family and all those who knew and worked with him."
Jetman Dubai also said the company is working closely with "all relevant authorities."
Reffet came from a family of skydivers and was part of the "Jetman" team which performed groundbreaking stunts above the emirate using jetpacks and carbon-fibre wings.
The Jetmen have pulled off a series of dramatic flights, soaring in tandem above the Burj Khalifa, and alongside an Emirates Airbus A380, the world's largest commercial airliner.
Fans are also sending messages describing the wingsuit jumper as a legend and a hero.
On the occasion of Union Pledge Day, UAE President His Highness Sheikh Mohamed bin Zayed Al Nahyan has reaffirmed the nation's dedication to unity, progress and honouring the legacy of its founding father, the late Sheikh Zayed bin Sultan Al Nahyan.
UAE President His Highness Sheikh Mohamed bin Zayed Al Nahyan and Hungarian Prime Minister Viktor Orbán witnessed the signing of agreements covering areas including AI, energy and defence cooperation.
Foreign ministers from 11 Arab nations, including the UAE, have expressed unified support for Syria’s sovereignty while strongly condemning the recent Israeli attacks, calling them a violation of international law.
The UAE has expressed its condolences and solidarity with South Korea following severe floods and torrential rains that resulted in multiple fatalities.