We present a novel flow instability that can arise in thin films of cytoskeletal fluids if the friction with the substrate on which the film lies is sufficiently strong. We consider a two-dimensional, membrane-bound fragment containing actin filaments that polymerize at the edge and depolymerize in the fragment. Performing a linear stability analysis of the initial state due to perturbations of the fragment boundary, we find, in the limit of large friction, that the perturbed actin velocity and pressure fields obey the same laws governing the viscous fingering instability of an interface between immiscible fluids in a Hele-Shaw cell. A remarkable feature of this instability is that it is independent of the strength of the interaction between actin filaments and myosin motors.