Is a non-euclidean-norm preserving map necessarily linear? Let $V$ and $W$ be two normed vector spaces and let $f:V \rightarrow W$ be a norm preserving map. I know that if both norms correspond to some inner product then $f$ is necessarily linear, but I can't find the answer for the more general case of normed vector spaces.
I suspect the answer is no, so I tried to come up with a counter-example involving "pseudo" rotations along non-euclideanly-spherical paths centered at the origin of $\mathbb R^2$, unsuccessfully.
I'd most importantly like an answer that does not assume $f$ to be surjective. However, any additional information about that particular case would be appreciated as well.

Take the map $\mathbb{R} \to \mathbb{R}, r \mapsto |r|$. This certainly preserves the Euclidean norm, but is not even additive, so not linear.
Another kind of dumb example is the trivial norm taking $0$ to $0$ and anything else to $1$: then any map taking $0_V$ to $0_W$ and $V \setminus\{0\}$ to $W \setminus\{0\}$ is norm preserving.
Or, you could take $V$ to have the trivial norm, and $W$ to be $\mathbb{C}$ with the Euclidean norm. Then any map $V \to W$ which maps $0$ to $0$ and maps $V \setminus \{0\}$ onto the unit circle is norm preserving. This certainly isn't surjective.
