Dillinger saw a woman he liked, Billie Frechette, played by Marion Cotillard, and courted her, after his fashion. That is, he took her out at night and bought her a fur coat, as he had seen done in the movies; he had no real adult experience before prison. They had sex, but the movie is not much interested. It is all about his vow to show up for her, to protect her. Against what? Against the danger of being his girl. He allows himself a tiny smile when he gives her the coat, and it is the only vulnerability he shows in the movie.