  Short $\mathbb{C}^2$'s were constructed in [F] as attracting basins of a
sequence of holomorphic automorphisms whose rate of attraction increases
superexponentially. The goal of this paper is to show that such domains also
arise naturally as autonomous attracting basins: we construct a transcendental
H\'enon map with an oscillating wandering Fatou component that is a Short
$\mathbb{C}^2$. The superexponential rate of attraction is not obtained at
single iterations, but along consecutive oscillations.
