The Ukrainian Defence Forces do not believe pilot error was behind the incident, the source added.
The pilot, Oleksiy Mes known as "Moonfish," was killed in the crash while "repelling the biggest ever aerial attack" by Russia against Ukraine, said the source, adding that the pilot was buried on Thursday.
The crash is being investigated and international experts will be invited to participate in the probe, the source added.
The death of the pilot is a major blow for Ukraine. The first F-16s only arrived in the country earlier this month and Moonfish was one of the few pilots trained to fly them.
Zelensky said on Tuesday that the Ukrainian Air Force used the F-16 to destroy missiles and drones launched by Russia on Monday, the first time any Ukrainian official confirmed the jets were being used in combat.
Kyiv waited a long time to get hold of the F-16s, and the country's President Volodymyr Zelensky has been asking its Western allies for the fighter jets since the start of the full-scale invasion.
But as with other equipment, Western countries hesitated before finally agreeing to provide F-16s. The Netherlands and Denmark pledged to provide them early summer in 2023, but it took another few weeks for the US to green-light the transfer.
When they arrived earlier this summer, Zelensky said he and his government held "hundreds of meetings and negotiations" to secure the jets.
A group of Ukrainian pilots started their training in the US in the fall. While it normally takes a year to get fully trained up to fly the planes, Moonfish and others had to do it in six months.
Moonfish: Top pilot who lobbied hard for F-16
Moonfish and another pilot Andriy Pilshchikov, known by the call sign "Juice," became the faces of Ukraine's campaign to get the F-16s.
It was an uphill battle, but Juice and Moonfish pulled through it together. They were young and enthusiastic, spoke good English and were willing to fight to get the US jets into Ukrainian skies.
Flying the F-16 was their dream and when Juice died in a plane crash during a combat mission last August, Moonfish made it his goal to fulfill it.
Of the two, Moonfish was the quiet one: an aviation geek not keen on publicity. But when Juice died, Moonfish had to take his place. In a rare interview with CNN, he admitted that if Juice were alive, he'd be giving interviews.
A man of few words, passionate about his job, he had his emotions under control. A straight talker who knew everything about F-16s.
"Andriy was the 'ideas man' and the main driving force behind it all," Moonfish said. "And I feel responsible to him for ensuring these planes arrive."
Speaking to CNN while in training, he said it was necessary for him and other Ukrainian pilots to undergo a truncated version of the training. "We would have had a lot of time to study the jet completely in peacetime, but we do not have the time," he said.
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