The structural relaxation times of metallic glasses (MGs) are generally believed to decrease with increasing flow rate, also known as shear thinning. In this work, an extraordinary loading rate-independent creep relaxation time, in contrast to the decreasing Maxwell relaxation time with increasing loading rate, is observed for a La-based MG with pronounced Johari–Goldstein relaxation under nanoindentation. The Johari–Goldstein relaxation in the flow of MGs is suggested to be a form of local Newtonian flow and may be not crucial in shear thinning, whereas the main α relaxation is proposed to predominate in the shear thinning of MGs, based on the disparity between the creep relaxation time and the Maxwell relaxation time. These results improve the current understandings on the structural relaxation and its role in the shear thinning and plastic deformation of MGs.