On a metric measure space (M,ρ,μ) we consider a family of Lipschitz–Besov spaces Λsp,q that is defined only using the metric ρ and measure μ, and a family of Besov spaces Bsp,q that is defined using an auxiliary self-adjoint operator L and the associated heat semigroup. Under certain assumptions about the heat kernel of L, we prove the identity of the two families of the function spaces.