Fatou-Bieberbach domains in $\mathbb C^n\setminus\mathbb R^k$

We construct Fatou-Bieberbach domains in $\mathbb C^n$ for $n>1$ which contain a given compact set $K$ and at the same time avoid a totally real affine subspace $L$ of dimension $<n$, provided that $K\cup L$ is polynomially convex. By using this result, we show that the domain $\mathbb C^n\setminus\mathbb R^k$ for $1\le k<n$ enjoys the Oka property with approximation for maps from any Stein manifold of dimension $<n$.