  In this paper we prove a rigidity statement for free boundary minimal
surfaces produced via min-max methods. More precisely, we prove that for any
Riemannian metric $g$ on the 3-ball $B$ with non-negative Ricci curvature and
$\mathrm{II}_{\partial B}\ge g_{|\partial B}$, there exists a free boundary
minimal disk $\Delta$ of least area among all free boundary minimal disks in
$(B,g)$. Moreover, the area of any such $\Delta$ equals to the width of
$(B,g)$, $\Delta$ has index one, and the length of $\partial\Delta$ is bounded
from above by $2\pi$. Furthermore, the length of $\partial\Delta$ equals to
$2\pi$ if and only if $(B,g)$ is isometric to the Euclidean unit ball. This is
related to a rigidity result obtained by F.C. Marques and A. Neves in the
closed case. The proof uses a rigidity statement concerning half-balls with
non-negative Ricci curvature which is true in any dimension.
