The Fermi surface that characterizes the electronic band structure of crystalline solids can be difficult to image experimentally in a way that reveals local variations. We show that Fermi surfaces can be imaged in real space with a low-temperature scanning tunneling microscope when subsurface point scatterers are present: in this case, cobalt impurities under a copper surface. Even the very simple Fermi surface of copper causes strongly anisotropic propagation characteristics of bulk electrons that are confined in beamlike paths on the nanoscale. The induced charge density oscillations on the nearby surface can be used for mapping buried defects and interfaces and some of their properties.