Theoretical calculations, computer simulations, and experiments indicate the possible existence of a ferromagnetic liquid state, although definitive experimental evidence is lacking. Should such a state exist, demagnetization effects would force a nontrivial magnetization texture. Since liquid droplets are deformable, the droplet shape is coupled with the magnetization texture. In a thin-film geometry in zero applied field, the droplet has a circular shape and a rotating magnetization texture with a point vortex at the center. We calculate the elongation and magnetization texture of such a ferromagnetic thin-film liquid droplet confined between two parallel plates under 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 films.