We report electron spin resonance (ESR) measurements on stage-I potassium intercalated graphite (KC8). Angular dependent measurements show that the spin-lattice relaxation time is longer when the magnetic field is perpendicular to the graphene layer as compared to when the magnetic field is in the plane. This anisotropy is analyzed in the framework of the Elliott-Yafet theory of spin-relaxation in metals. The analysis considers an anisotropic spin-orbit Hamiltonian and the first order perturbative treatment of Elliott is reproduced for this model Hamiltonian. The result provides an experimental input for the first-principles theories of spin-orbit interaction in layered carbon and thus to a better understanding of spin-relaxation phenomena in graphene and in other layered materials as well.