In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let ((M)n+1,(F)) be a Randers–Minkowski space. If (Mn,F) is a hypersurface of ((M)n+1,(F)) with constant flag curvature K = 1, then we can prove that M is Riemannian. (2) Let ((M)n+1,(F)) be a Randers space with constant flag curvature. Assume (M,F) is a compact hypersurface of ((M)n+1,(F)) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of [Proc. Amer. Math. Soc., 120, 1223–1229 (1994)] from the Riemannian case to the Randers space.