Scanning probe microscopes are widely used to study surfaces with atomic resolution in many areas of nanoscience. Ultracold atomic gases trapped in electromagnetic potentials can be used to study electromagnetic interactions between the atoms and nearby surfaces in chip-based systems. Here we demonstrate a new type of scanning probe microscope that combines these two areas of research by using an ultracold gas as the tip in a scanning probe microscope. This cold-atom scanning probe microscope offers a large scanning volume, an ultrasoft tip of well-defined shape and high purity, and sensitivity to electromagnetic forces (including dispersion forces near nanostructured surfaces). We use the cold-atom scanning probe microscope to non-destructively measure the position and height of carbon nanotube structures and individual free-standing nanotubes. Cooling the atoms in the gas to form a Bose-Einstein condensate increases the resolution of the device.