One of the major goals of research for laser-plasma accelerators1 is the realization of compact sources of femtosecond X-rays2,3,4. In particular, using the modest electron energies obtained with existing laser systems, Compton scattering a photon beam off a relativistic electron bunch has been proposed as a source of high-energy and high-brightness photons. However, laser-plasma based approaches to Compton scattering have not, to date, produced X-rays above 1 keV. Here, we present a simple and compact scheme for a Compton source based on the combination of a laser-plasma accelerator and a plasma mirror. This approach is used to produce a broadband spectrum of X-rays extending up to hundreds of keV and with a 10,000-fold increase in brightness over Compton X-ray sources based on conventional accelerators5,6. We anticipate that this technique will lead to compact, high-repetition-rate sources of ultrafast (femtosecond), tunable (X- through gamma-ray) and low-divergence (∼1°) photons from source sizes on the order of a micrometre. Scientists demonstrate a Compton-based electromagnetic source based on a laser-plasma accelerator and a plasma mirror. The source generates a broadband spectrum of X-rays and is 10,000 times brighter than Compton X-ray sources based on conventional accelerators.