  We establish Marstrand-type as well as Besicovich-Federer-type projection
theorems for closest-point projections onto hyperplanes in the normed space
$\mathbb{R}^{n}$. In particular, we prove that if a norm on $\mathbb{R}^{n}$ is
$C^{1,1}$-regular, then the analogues of the well-known statements from the
Euclidean setting hold. On the other hand, we construct an example of a
$C^{1}$-regular norm in $\mathbb{R}^2$ for which Marstrand-type theorems fail.
