  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 $<n$.
