The concept of gyrocommutative gyrogroups is defined by Ungar. It is a generalization of the concept of commutative groups. A gyrocommutative gyrogroup is not necessarily commutative nor associative. A concrete example of a gyrocommutative gyrogroup is provided by admissible velocities in Einstein's special relativity, and another concrete example is provided by the Poincaré disk model of hyperbolic geometry. In this talk, we consider a generalization of real normed spaces based on gyrocommutative gyrogroups and give a Mazur-Ulam type theorem for such spaces.