An unmanned aerial vehicle is an aerial vehicle, so... the Kongming lantern is famously attributed to Zhuge Liang (courtesy name Kongming), from the second century AD, who used it to send a signal to military allies. Wikipedia's article cites a historian, sans citation, claiming much earlier usage. I'll leave it to a more motivated reader to find a reference to this that doesn't trace back to Wikipedia itself - possibly this has something? (Laments for Aaron)
Since the principle of the sky lantern scales up to hot air balloons, there is no telling what attempts might have been lost to history, though one would surmise they did not work out well.
For now no, because we have not found something that could be solely a flying vehicle beyond doubt. But we have found tons of stuff, from coins, weapons and graves to entire building (temples, pyramids). But we have found nothing directing to a a flying vehicle.
The documentary mixes different archaeological devices to produce this assumption: I do the same with another real archaeological device:
For example we have found Antikythera mechanism: "The Antikythera mechanism (/ˌæntɪkɪˈθɪərə/, /ˌæntɪˈkɪθərə/) is an ancient Greek analogue computer used to predict astronomical positions and eclipses for calendar and astrological purposes decades in advance. It could also be used to track the four-year cycle of athletic games which was similar to an Olympiad, the cycle of the ancient Olympic Games."
We know that this artifact has a certain age, does a certain function but we do not know where it was used and why. It is certainly an amazing construction but we cannot even be sure if its functioning was accepted by the people of the time.
Assuming it was crafted by aliens instead of human minds of the time is equal to say that every important person on earth was actually either alien or under alien 'control'.
Similarly say that this device is part of an ancient spaceship build by humans with help from aliens and not a stand alone device counting something needed or just proving something, is just like above. We assume the most least possible against the most possible.
This answer assumes that 'ancient flying vehicle' will be something aged BC, relatively controllable, carry at least 1 person and 'fly' at a logical height of 1 meter or more and for some logical time of 1 minute or more. Thus i do not consider tower or mountain jumps of people wearing various sorts of 'wings' as 'flying vehicle'. The first 'flying vehicles' to fit those requirements where the well known balloons appearing about 18th century.