We report on the suppression of the Rayleigh-Plateau instability in films of giant micelles solutions coating a vertical fibre. We observed that the dynamics of the thin films coating the fibre could be very different from the Newtonian or standard Non-Newtonian cases. Focusing on the competition between surface tension and the elasticity of shear-induced structures well-known to develop in sheared micellar systems, we show that the films could be stabilized if two conditions are fulfilled. First, the shear-induced structures must appear in the flowing solution. Second, the elastic modulus of the shear-induced structures must be greater than a critical value that can be evaluated from a linear stability analysis, providing a way of estimating the bulk elasticity of these structures.