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We show that a thin film with small dynamic contact angle and driven by an external body force is unstable to the formation of fingers in the direction perpendicular to the main flow. The instability is largest in the capillary region near the contact line, where the force due to surface tension is comparable to the viscous and gravitational forces. The fastest growing wavelength is calculated in the limit of small-amplitude disturbances. These instabilities may be related to finger patterns observed in gravitational flows and spinning drops.