The incident occurred in Dubai on Tuesday morning local time, according to a statement from the company, posted on Facebook.
“Vince was a talented athlete, and a much-loved and respected member of our team,” the statement reads. “Our thoughts and prayers are with his family and all those who knew and worked with him.”
Jetman Dubai has developed a jet-powered wingsuit capable of taking off from the ground and then transitioning to high-altitude flight.
In February, Reffet, whose stunts using the jetpack have gone viral on social media, was the first pilot to demonstrate the transition to high-altitude flight.
Jetman pilots had previously launched from elevated platforms, such as a helicopter. Reffet was the first to launch from the ground.
The company said it announced Reffet’s death with “unimaginable sadness.”
“We are working closely with all relevant authorities and ask that you please keep Vince’s family in your thoughts and prayers,” it continued.
Jetman Dubai’s carbon-fiber wingsuit is powered by four mini jet engines, and the team’s engineers were able to create a manually controlled thrust vectoring nozzle that allows pilots “to control rotations around the yaw axis at zero speeds.”
The yaw axis is perpendicular to the wings and allows the pilot to turn left and right while flying horizontally.
Pilots can hover, stop, turn and maneuver, even performing rolls and loops.
The suit has a maximum altitude of 6,100 meters (20,000 feet), according to the Jetman Dubai website, a maximum flight duration of 13 minutes and a top speed of 220 knots (253 miles per hour).
In 2015, Reffet and fellow pilot Yves Rossy flew alongside the world’s largest passenger jet, an Airbus A380, over Dubai.
Reffet, aged 36, was also known for BASE jumping, and jumped from the Burj Khalifa, the world’s tallest building, in 2014.
CNN’s Allen Kim contributed to this report.