When a sphere settles through the free surface of a viscous fluid, the interface is deformed and assumes a funnel shape behind the sphere. If the fluid is viscoelastic and the settling process is fast compared to the relaxation time of the fluid, elastic effects are dominant and an instability occurs. The interface loses its original axisymmetry and buckles, leading to a particular mode of pinch-off unseen in Newtonian fluids. We present experimental evidence that stress boundary layers form in this type of flow, and argue that a physical mechanism for this instability can be recovered, at least qualitatively, by considering the stability of a stretched anisotropic elastic membrane in a pressure field.