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Electromagnetic fields not only induce electronic transitions but also fundamentally modify the quantum states of matter through strong light-matter interactions. As one established route, Floquet engineering provides a powerful framework to dress electronic states with time-periodic fields, giving rise to quasi-stationary Floquet states. With increasing field strength, non-perturbative responses of the dressed states emerge, yet their nonlinear dynamics remain challenging to interpret. In this work we explore the emergence of non-adiabatic Landau-Zener transitions among Floquet states in Cu(111) under intense optical fields. At increasing field strength, we observe a transition from perturbative dressing to a regime where Floquet states undergo non-adiabatic tunneling, revealing a breakdown of adiabatic Floquet evolution. These insights are obtained through interferometrically time-resolved multi-photon photoemission spectroscopy, which serves as a sensitive probe of transient Floquet state dynamics. Numerical simulations and the theory of instantaneous Floquet states allow us to directly examine real-time excitation pathways in this non-perturbative photoemission regime. Our results establish a direct connection the onset of light-dressing of matter, non-perturbative ultrafast lightwave electronics, and high-optical-harmonic generation in the solids.