Creeping waves play an important role in diffraction by a smooth convex body and give an asymptotics of the diffracted field in the shadow. Known results obtained by the boundary-layer method do not allow us to explain some of the properties of creeping waves on highly elongated bodies. In this paper, creeping waves on highly elongated bodies are studied in the case where the binormal curvature of the surface is asymptotically large. The asymptotics derived contains solutions of a differential equation of the Heun type. The analysis of the dispersion equation for the surface waves is carried out numerically. It is discovered that the magnetic creeping wave travels along the surface of a highly elongated body with much less attenuation than predicated by the usual theory.