Hemingway was well known for his love of cats, and indeed of nature in general. In fact, the polydactyl (six-toed) cat is often referred to as the 'Hemingway cat' due to the large numbers he kept at both his Key West home and the Finca Vigia (Lookout Farm) in Cuba.
F. Puss was a fluffy Persian bestowed on Bumby by Hadley's friend, Kitty Carnnell. According to Carlene Fredericka Brennen and Hilary Hemingway in Hemingway's Cats: An Illustrated Biography (p. 159) -- and as Hemingway goes on to corroborate later in A Moveable Feast -- 'F. Puss would babysit Bumby in Paris when the femme de ménage [charwoman/cleaning lady] was away and would not allow anyone to come near the boy.'
In the book's foreword, Hilary Hemingway quotes a song that Hemingway composed for the same feline: '"A feather kitty’s talent lies In scratching out the other’s eyes. A feather kitty never dies Oh immortality."'