When a dry sphere sinks into a fluid, a funnel-shaped free surface develops behind the sphere if the sinking occurs faster than the surface wetting. If the fluid is viscoelastic, the interface can become unstable to a loss of axisymmetry. The stress near this surface concentrates into boundary layers, as also seen in other free surface extensional flows of viscoelastic fluids. At high Deborah number and low Reynolds number, the qualitative behaviour can be recovered by considering the static equilibrium of a stretched elastic membrane in an hydrostatic pressure field. We treat this problem in the framework of finite elasticity using a neo-Hookean constitutive model, and show how the conditions of instability can be recovered. A numerical study of this model is presented.