The map which you call "geodesic preserving" is usually called "affine". It seems that affine maps to the real line are well understood even for general length space.
For your later edit: you may always take two spaces which admit self-similar maps and consider map on the product which move each coordinate with different coefficients.
So it is better to ask
A. Lytchak says that the following is there any irreducible (i.e. one which a well known open question:
If such map exist then the space can not be presented as a embedded into product ) CAT(0)-space which admits a non-self-similar affine maps to itself?
The right person to ask this question is A. Lytchakof spaces and the map preserves product structure.
P.S. Sorry the The "example" I gave before was not an example.
