  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.
