Before going out to the park, Miss Brill puts on a fur collar, which likely has the head of a small animal, like a fox or mink, still attached. Here, Miss Brill personifies the animal:
"She had taken it out of its box that afternoon, shaken out the moth-powder, given it a good brush, and rubbed the life back into the dim little eyes. ‘What has been happening to me?’ said the sad little eyes. Oh, how sweet it was to see them snap at her again..."
In this passage, Miss Brill not only imagines that the animal is still alive, but endows it with human characteristics and emotions. In her mind, the animal's eyes express sadness but also "snap at her" in a playful manner. Miss Brill even fantasizes that the animal can speak, and that it is both friendly towards and dependent on her.
By personifying the animal, Miss Brill illustrates the kind of mutually affectionate relationships she doesn't have with other humans. The people she encounters in the park are indifferent or actively hostile towards her; no one needs her as she imagines the animal on her collar does. The personification of the fur is one of several methods Mansfield uses to show Miss Brill's deep loneliness without explicitly naming it.