While I was researching my post on Tissandier’s ill-fated 1875 high-altitude balloon ride, I happened to come across a very curious image, pictured below.

Apparently the 1870 Franco-Prussian War not only resulted in the first airmail: it also spawned the very first aerial combat! As described in the 1902 book Travels in Space, by E.S. Valentine and F.L. Tomlinson,
During the war an incident of great dramatic interest is narrated as having occurred in full view of Paris. A French war-balloon, the “lntrepide,” was floating in the air 3,ooo metres above the fort at Charenton. Almost at the same time a second balloon, also flying the French colours, was observed on the horizon. When within a short distance of each other, a loud report was heard high in the heavens, followed by a series of explosions.
“The voices,” writes an eye-witness, “were at first thought by the cheering garrison below to be demonstrations or signals of victory, until one aeronaut was seen to fling himself into the network of his balloon and to cling to its sides. During this time the other continued discharging shots which were traced in the sky by the luminous effects.
“The ‘Intrepide descended rapidly, and it appeared to the spectators below that some incomprehensible event had taken place above. Suddenly the French flag of the second balloon was removed, and a black and yellow standard was perceived to be floating in its place.

Flag of Prussia from 1803-1892. Via Wikipedia.
The cry went up ‘Treason ‘-it is a Prussian balloon ! The Prussian balloonist has fired on the ‘ Intrepide’! The ‘Intrepide’ was, however, safe, for her aeronaut was seen to descend rapidly in his car and the balloon nearly to reach the earth. He cast out the ballast and re-ascended, having hastily closed the hole made in his balloon by his adversary. Shots were instantly fired from the ‘Intrepide’ into the Prussian balloon, which, losing all power, fell with terrific velocity. A detachment of Uhlans who were in the plain and had been following the course of the exciting aerial combat, rushed forward and surrounded their champion. He was carried off injured, but how great were his injuries none ever knew.”
Was this incident the very first air combat? It’s hard to imagine any battles happening earlier. It is hard to imagine that this was a common occurrence in the history of ballooning — considering the complete lack of control in horizontal motion, it would be extremely uncommon for two lighter-than-air craft to get close enough for a fight.
Update: See comment below by Mike Monaco!