This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichm"uller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichm"uller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, which includes strictly convex bounded domains, as well as Teichm"uller spaces.