Drake, 33, passed away on Wednesday evening in hospital, Bern cantonal police said in a statement.
Police did not reveal the victim's name, but on its Facebook page the World Wingsuit League (WWL) identified him as Drake, saying he had been in a coma since Saturday's accident.
The league set up a special page in Drake's honour, where dozens of people posted messages and shared memories, photos and videos of his jumps and footage he shot of others in action.
"He was the most incredible, wise, kind, adventurous, funny, supportive, loving brother one could ever wish for," wrote his younger brother, Jordan Drake, in a moving tribute on the site.
Wingsuit divers wear special jumpsuits that add surface area to the body, enabling them to glide like a bird, before opening a parachute like a regular skydiver.
Drake was one of the biggest names in the wingsuit community, travelling the globe for competition meets and known for his breathtaking jumps.
Along with experienced wingsuit flyers New Zealander Dan Vicary, 33, and French wingsuit icon Ludovic Woerth, 34, Drake had jumped from a helicopter over the Lütschental near Bern last Saturday.
The trio had planned to land in the valley, but crashed into an Alpine pasture, police said.
Vicary and Woerth were found dead on site when rescuers arrived at the scene, and the American was rushed to a hospital by helicopter.
The investigation into the causes of the accident is ongoing, police said.
The wingsuit league offered its condolences to the family of the sportsman, who had moved from the United States to Europe to practise his sport.
The Swiss Alps are a magnet for wingsuit divers, base-jumpers and other extreme sports enthusiasts.
On its website, the Swiss newspaper Blick published an interview with French wingsuit diver Vincent Descols, who had made the same jump the previous day.
Descols told Blick that the cause of the accident appeared to be a dive-zone error after the three men jumped from the helicopter at an altitude of 2,750 metres (9,022 feet).
But instead of finding themselves above a deep gorge, as planned, they were actually above the flat pasture.
Unable to reach the safety of the gorge, they had just three seconds to open their parachutes, which was too little time.
"I'm really upset, but I'm not blaming anyone," Descols told Blick, saying that nobody would know how to react in such a high-pressure situation.
"All three of them were great flyers, and this could happen to any of us," he said.
Around 20 people are killed each year taking part in the extreme sport.
British wingsuit diver and stuntman Mark Sutton— best known for parachuting into the London 2012 Olympics opening ceremony dressed as James Bond — died last August when he crashed into a Swiss mountain ridge after jumping from a helicopter.
This YouTube video from WWL, highlighting the exploits of another American wingsuit flyer, Jeb Corliss, illustrates the dangers involved: