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.
UAE President His Highness Sheikh Mohamed bin Zayed Al Nahyan has held a phone call with the President of the European Commission, Ursula von der Leyen, to discuss ways to strengthen ties in support of shared interests.
UAE President His Highness Sheikh Mohamed bin Zayed Al Nahyan has discussed enhancing cooperation with Niko Christodoulides, President of Cyprus, as part of the Comprehensive Strategic Partnership between the two nations.
Emirates Airline has set up an airbridge with Dubai Humanitarian to transport urgent relief aid to support those affected by the devastating Cyclone Ditwah in Sri Lanka.
Road and Transport Authority (RTA) has outlined the route map for Dubai Metro's Blue Line that will connect 14 stations, of which five will be elevated, four underground, two "future elevated" and one underground interchange.
The UAE has inaugurated the Emirates Medical Centre in Khan Younis, south of the Gaza Strip, that aims to support the health sector across the territory by delivering urgent medical care.
UAE President His Highness Sheikh Mohamed bin Zayed Al Nahyan has started his official visit to Cyprus with a reception ceremony held by President of Cyprus, Nikos Christodoulides, at the Presidential Palace in Nicosia.
Several parts of the UAE have been hit with rainfall of varying intensities on Sunday morning, with the Dubai Police issuing public safety messages via SMS to urge residents to exercise caution.
The UAE has strongly condemned the terrorist shooting at a Jewish gathering in Australia's Bondi Beach that left at least 12 people dead and around 30 more wounded.