We study deformations of the "Very Special Relativity" model of Lorentz violation, introduced by Cohen and Glashow. Unlike in ordinary special relativity, we find there is no deformation that could be interpreted as the passage to a curved gravitational background, but there does exist a non-trivial deformation, which could be induced through quantum effects. It implies a modification from ordinary Minkowskian spacetime geometry to a Finslerian geometry. Experimental bounds on the deformation parameter are very stringent, which poses the theoretical puzzle, analogous to that for the cosmological constant in standard relativity, of why the deformation parameter is so small.