Got me curioser, so I googled it. One source said what I thought:
https://www.scientificamerican...
"Because gravity is necessary for density differences to arise, neither buoyancy nor convection occur in a zero-gravity environment such as space. Consequently, the combustion products accumulate around the flame, preventing sufficient oxygen from reaching it and sustaining the combustion reaction. Ultimately the flame goes out."
and
"Researchers learned that flames extinguish themselves."
and
"Oxygen could still reach a flame in a gravity-free environment if someone blew the gas into the flame or let it "diffuse" in. It is the diffusion process that spreads the scent of a perfume in a room without air circulation: the perfume slowly mixes with the air to try to achieve a uniform distribution. This process, however, is too slow to sustain a flame."
Other sites don't directly contradict this, but say fires in the ISS are dangerous because smoke doesn't rise and set off smoke detectors on ceilings like in homes, so they install smoke detectors in the ventilation ducts. Also that fires on the ISS can survive on lower levels of oxygen than humans, and thus are much more dangerous if they linger on. That's confusing; if the smoke doesn't rise, then wouldn't it smother the fire like the first site says? But if the ISS has moving air from ventilation ducts, maybe that is what feeds oxygen to the fires.
Thanks for tricking me into not being so lazy :-) But all I learned is that I don't know.