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.
Sharjah Police have arrested a motorist who racked up 137 traffic violations and fines totalling over AED 104,000, all while using forged licence plates to evade detection.
Economic and developmental cooperation topped the agenda during discussions between His Highness Sheikh Abdullah bin Zayed Al Nahyan, Deputy Prime Minister and Minister of Foreign Affairs, and President of Ecuador, Daniel Noboa.
The UAE and Lebanon have agreed to enhance cooperation in key sectors, including economic and investment, "through joint opportunities, knowledge exchange and government development".
The UAE's President His Highness Sheikh Mohamed bin Zayed Al Nahyan and his Russian counterpart Vladimir Putin have held a phone call to discuss ways to strengthen ties under their countries’ strategic partnership.
His Highness Sheikh Mohammed bin Rashid Al Maktoum, Vice President and Prime Minister of the UAE and Ruler of Dubai, toured the Arabian Travel Market (ATM), which wrapped up on Thursday at the Dubai World Trade Centre.