Short answer: yes, but...
There are already a number of good answers and examples that, while unprobable (in the sense of "requiring a specific combination of factors") and dangerous in most of the cases, it is definitely really possible (since it happened).
Adding an historical example
At the beginning of commercial aviation, unreliable planes carried paper mail (a.k.a. post, letters) in daredevil conditions.
In between 1927 and 1929, Jean Mermoz was trying to open a route across the Andes. He and his mechanic stranded at high altitude after an incident, rode into a precipice to gain enough speed to fly again. This has been widely documented for decades.
See for example:
Jean Mermoz (1901 - 1936) - SP's Aviation
He rode an updraft that carried the plane through a high mountain
pass, but then a downdraft slammed the aircraft onto a plateau at
12,000 feet. Though the small machine suffered only mild damage and
was still airworthy, there wasn’t enough take-off run available on the
plateau. Mermoz cleared a rough path to the edge of the precipice and
they removed whatever they could from the aircraft. They strapped
themselves in and Mermoz just rolled off the mountain hoping to gain
enough flying speed before hitting the ground. They succeeded;
otherwise it would have been certain death.
Famous Aviators | 10 famous aviators
Mermoz next tackled shortening the Argentina-to-Chile route; pilots had to make a thousand-mile detour to get around the Andes. With mechanic Alexandre Collenot, Mermoz set out in a Latécoère 25 monoplane and found an updraft that carried them through a mountain pass, but a downdraft smashed the aircraft onto a plateau at 12,000 feet. After determining that they could not hike out, Mermoz cleared a crude path to the edge of the precipice and removed from the aircraft anything that wasn’t bolted down. He and Collenot strapped themselves in, and Mermoz got the airplane rolling down the path.
In effect, they dove off the mountain, and Mermoz pointed the nose straight down, hoping to gain flying speed. Again, luck was with him.