  Recent theoretical analysis of spatially-nonuniform modes of the
thermomagnetic instability in superconductors [Phys. Rev. B 70, 224502 (2004)]
is generalized to the case of a thin film in a perpendicular applied field. We
solve the thermal diffusion and Maxwell equations taking into account nonlocal
electrodynamics in the film and its thermal coupling to the substrate. The
instability is found to develop in a nonuniform, fingering pattern if the
background electric field, E, is high and the heat transfer coefficient to the
substrate, h0, is small. Otherwise, the instability develops in a uniform
manner. We find the threshold magnetic field, H_fing(E,h0), the characteristic
finger width, and the instability build-up time. Thin films are found to be
much more unstable than bulk superconductors, and have a stronger tendency for
formation of dendritic pattern.
