Electrodynamical processes induced in complex systems like semiconductors by strong electromagnetic fields have traditionally been described using semiclassical approaches. Although these approaches allowed the investigation of ultrafast dynamics in solids culminating in multipetahertz electronics, they do not provide any access to the quantum-optical nature of the interaction, as they treat the driving field classically and unaffected by the interaction. Here, using a full quantum-optical approach, we demonstrate that the subcycle electronic response in a strongly driven semiconductor crystal is imprinted in the quantum state of the driving field resulting in nonclassical light states carrying the information of the interaction. This vital step towards strong-field ultrafast quantum electrodynamics unravels information inaccessible by conventional approaches and leads to the development of a new class of nonclassical light sources.