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.
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 contract has been awarded for a project that will see the development of Sheikh Zayed bin Hamdan Al Nahyan Street Intersection with Al Awir Road and Al Manama Street to reduce travel time from 20 minutes to five.
The UAE Relief Team has departed Sri Lanka after completing its humanitarian mission, carried out in response to the severe impact of Cyclone Ditwah and the landslides that struck the country.
His Highness Sheikh Abdullah bin Zayed Al Nahyan, UAE Deputy Prime Minister and Minister of Foreign Affairs, and Serbia’s Foreign Minister Marko Đurić, have agreed to deepen political coordination after signing a Memorandum of Understanding (MoU) during talks in Abu Dhabi.
The UAE has introduced changes to its laws governing the care of children of unknown parentage, expanding custody options and strengthening child protection measures.
UAE-backed medical teams have launched a series of humanitarian missions in rural Pakistan, providing free treatment for children and elderly patients with heart conditions.
Dubai authorities have begun final preparations for the New Year celebrations, holding a coordination meeting to review security and operational plans across the emirate.
The UAE's National Centre of Meteorology (NCM) says a surface low-pressure system will affect the country from December 13 to 19, bringing periods of instability.