In this paper, we show that n-dimensional (n ≥ 2) complete and non-compact smooth metric measure spaces with non-negative weighted Ricci curvature in which some Gagliardo-Nirenberg-type inequality holds are not far from the model metric measure n-space (i.e., the Euclidean metric n-space). Moreover, this fact, together with two generalized volume comparison theorems given in [P. Freitas et al. Calc. Var. Partial Differential Equations 51 (2014), 701-724], surprisingly leads to an interesting rigidity theorem for the given metric measure spaces.