abstract: I will present the construction of a family of automorphisms of $\mathbb{C}2$ having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb{C}\times\mathbb{C}$ and which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. (This is a joint work with Filippo Bracci and Berit Stensones).