Theoretical calculations, computer simulations, and experiments indicate th
e possible existence of a ferromagnetic liquid state, although definitive e
xperimental evidence is lacking. Should such a state exist, demagnetization
effects would force a nontrivial magnetization texture. Since liquid dropl
ets are deformable, the droplet shape is coupled with the magnetization tex
ture. In a thin-film geometry in zero applied field, the droplet has a circ
ular shape and a rotating magnetization texture with a point vortex at the
center. We calculate the elongation and magnetization texture of such a fer
romagnetic thin-film liquid droplet confined between two parallel plates un
der a weak applied magnetic field. The vortex stretches into a domain wall
and exchange forces break the reflection symmetry. This behavior contrasts
qualitatively and quantitatively with the elongation of paramagnetic thin f
ilms.