It is obtained as a byproduct of clay gathering. The chance is random, but about 10% of the time, collecting clay will also yield 1 flint.
I (ChildeRoland) picked over 200 clay. The first 100-120 gave me 11 flint, but the rest gave me no additional. I don't know if that is because there is some kind of limit, but laws of probability say that I should've gotten something in that last 80-100 clay. Could someone else investigate this? Later I gathered 52 Clay with only finding 2 Flint.
Aolen: I've noticed a similar trend. Clay beds in busy areas tend to have less flint that isolated patches. Perhaps flint regenerates slowly in a clay bed?
We are in an isolated area in Khartoum. Yesterday I got 6 flint for about 105 clay. Today in the same clay bed my partner picked up 63 flint for 656 clay.
Silly trivia item: If you have a good ear and listen under good headphones, you can perceive that the sound of gathering flint has the quality of being inside a small room rather than outside. --JulianJaynes