  We construct holomorphically varying families of Fatou-Bieberbach domains
with given centres in the complement of any compact polynomially convex subset
$K$ of $\mathbb C^n$ for $n>1$. This provides a simple proof of the recent
result of Yuta Kusakabe to the effect that the complement $\mathbb C^n\setminus
K$ of any polynomially convex subset $K$ of $\mathbb C^n$ is an Oka manifold.
The analogous result is obtained with $\mathbb C^n$ replaced by any Stein
manifold with the density property.
