High-power lasers have numerous scientific and industrial applications. Some key areas include laser cutting and welding in manufacturing, directed energy in fusion reactors or defense applications, laser surgery in medicine, and advanced photolithography in the semiconductor industry. These applications require optical components, in particular mirrors, that withstand high optical powers for directing light from the laser to the target. Ordinarily, mirrors are comprised of multilayer coatings of different refractive index and thickness. At high powers, imperfections in these layers lead to absorption of light, resulting in thermal stress and permanent damage to the mirror. Here we design, simulate, fabricate, and demonstrate monolithic and highly reflective dielectric mirrors which operate under high laser powers without damage. The mirrors are realized by etching nanostructures into the surface of single-crystal diamond, a material with exceptional optical and thermal properties. We measure reflectivities of greater than 98% and demonstrate damage-free operation using 10 kW of continuous-wave laser light at 1070 nm, with intensities up to 4.6 MW/cm2. In contrast, at these laser powers, we observe damage to a standard dielectric mirror based on optical coatings. Our results initiate a new category of broadband optics that operate in extreme conditions.