In the regime of deep strong light-matter coupling, the coupling strength exceeds the transition energies of the material1-3, fundamentally changing its properties4,5; for example, the ground state of the system contains virtual photons and the internal electromagnetic field gets redistributed by photon self-interaction1,6. So far, no electronic excitation of a material has shown such strong coupling to free-space photons. Here we show that three-dimensional crystals of plasmonic nanoparticles can realize deep strong coupling under ambient conditions, if the particles are ten times larger than the interparticle gaps. The experimental Rabi frequencies (1.9 to 3.3 electronvolts) of face-centred cubic crystals of gold nanoparticles with diameters between 25 and 60 nanometres exceed their plasmon energy by up to 180 per cent. We show that the continuum of photons and plasmons hybridizes into polaritons that violate the rotating-wave approximation. The coupling leads to a breakdown of the Purcell effect-the increase of radiative damping through light-matter coupling-and increases the radiative polariton lifetime. The results indicate that metallic and semiconducting nanoparticles can be used as building blocks for an entire class of materials with extreme light-matter interaction, which will find application in nonlinear optics, the search for cooperative effects and ground states, polariton chemistry and quantum technology4,5.