Strongly driven systems of emitters offer an attractive source of light over broad spectral ranges up to the X-ray region. A key limitation of these systems is that the light they emit is mostly classical. We overcome this constraint by building a quantum-optical theory of strongly driven many-body systems, showing that the presence of correlations among the emitters creates emission of non-classical many-photon states of light. We consider the example of high-harmonic generation, by which a strongly driven system emits photons at integer multiples of the drive frequency. In the conventional case of uncorrelated emitters, the harmonics are in an almost perfectly multi-mode coherent state lacking any correlation between harmonics. By contrast, a correlation of the emitters before the strong drive is converted into non-classical features of the output light, including doubly peaked photon statistics, ring-shaped Wigner functions and correlations between harmonics. We propose schemes for implementing these concepts, creating the correlations between emitters via an interaction between them or their joint interaction with the background electromagnetic field. Our work paves the way towards the engineering of novel states of light over a broadband spectrum and suggests high-harmonic generation as a tool for characterizing correlations in many-body systems with attosecond temporal resolution.