  Hujter and L\'angi introduced the $k$-fold Borsuk number of a set $S$ in
Euclidean $n$-space of diameter $d > 0$ as the smallest cardinality of a family
$\mathcal F$ of subsets of $S$, of diameters strictly less than $d$, such that
every point of $S$ belongs to at least $k$ members of $\mathcal F$.
  We investigate whether a $k$-fold Borsuk covering of a set $S$ in a finite
dimensional real normed space can be extended to a completion of $S$.
Furthermore, we determine the $k$-fold Borsuk number of sets in not angled
normed planes, and give a partial characterization for sets in angled planes.
