We report a surface instability observed during the extrusion of extremely soft elastic solids in confined geometries. Because of their unique rheological properties, these soft solids can migrate through narrow gaps by continuously everting the bulk material. The extrusion front spontaneously buckles in the direction transverse to the flow, resulting in a furrowlike morphology that deepens over time. We characterize the distinct features of this instability using experiments and theory and contrast the results with known elastic surface instabilities. Our Letter may provide insights into various processes involving the extrusionlike deformations of soft solids, including biomaterials.