You don’t need a ring, you need a conical group, that is a group with a one-parameter 
of contracting automorphisms. This type of generalization of affine geometry following 
Artin style has been done here: [Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91–136][1].


  [1]: https://arxiv.org/abs/0804.0135