Onibaba
(1964)

The "demon" is actually the mother (Nobuko Otowa), who has been wearing the mask she stole from the murdered samurai, to scare her daughter-in-law (Jitsuko Yoshimura) from visiting their neighbour Hachi (Kei Sato) every night.

After spending the night with Hachi in the grass, the daughter-in-law returns to their shack to find the mother wearing the mask which is now stuck to her face. They cannot remove the mask, so the daughter smashes it with a mallet. The mother's face is now scared and diseased, just like the face of the samurai she stole the mask from.

Convinced the mother is now a demon, the daughter-in-law flees, chased by the mother. (It is implied that the mother may fall into the pit where the duo threw the corpses of the warriors they killed).

When Hachi returns to his shack, he finds a wanderer eating his food. Hachi is killed and the man flees.

Thanks Ali D!