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.


H.H. Sheikh Hamdan pays tribute to royal photographer
H.H. Sheikh Hamdan awards Arab Hope Makers
UAE looks into framework to regulate children's social media use
UAE Ramadan moon-sighting committee to meet on Tuesday
RTA opens second bridge at Al Qudra intersection
Abu Dhabi identifies over 40 modern heritage sites
Distribution of aid from UAE's Saqr Humanitarian Ship begins in Gaza
Dubai to expand bus and taxi lanes as public transport ridership grows by 7.4%
Emir of Qatar welcomed by UAE President on visit to Abu Dhabi
UAE, Egypt Foreign Ministers discuss Gaza peace plan
UAE, Pakistan leaders discuss boosting economic ties
UAE, Bosnia FMs discuss strengthening bilateral relations
H.H. Sheikh Mohammed launches campaign to rescue 5 million children from hunger
Dubai advances measures to support people of determination
Abu Dhabi Crown Prince to lead UAE delegation at AI Impact Summit in India
Over 14 million Captagon pills, hidden in grain sacks, seized in UAE–Kuwait operation
Remote learning on Fridays during Ramadan for UAE public schools
UAE clarifies enrollment rules for British curriculum schools
UAE announces temporary road closures in Al Dhaid
Umm Al Emarat aid ship sets sail for Gaza Strip
