We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $Ω$ and the Busemann function associated with any straight line, the volume of the intersection of $Ω$ and any sublevel set of the Busemann function that contains a barycenter of $Ω$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.