  We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an
invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times
(\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits
converge to a fixed point on the boundary of the component. Such a Fatou
component also avoids $k$ analytic discs intersecting transversally at the
fixed point. As a corollary, we obtain a Runge copy of $\mathbb C \times
(\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.
