Extreme ultraviolet (EUV) high-harmonic radiation(1,2) emerging from laser-driven atoms, molecules or plasmas underlies powerful attosecond spectroscopy techniques(3-5) and provides insight into fundamental structural and dynamic properties of matter(6,7). The advancement of these spectroscopy techniques to study strong-field electron dynamics in condensed matter calls for the generation and manipulation of EUV radiation in bulk solids, but this capability has remained beyond the reach of optical sciences. Recent experiments(8,9) and theoretical predictions(10-12) paved the way to strong-field physics in solids by demonstrating the generation and optical control of deep ultraviolet radiation(8) in bulk semiconductors, driven by femtosecond mid-infrared fields or the coherent up-conversion of terahertz fields to multi-octave spectra in the mid-infrared and optical frequencies(9). Here we demonstrate that thin films of SiO2 exposed to intense, few-cycle to sub-cycle pulses give rise to wideband coherent EUV radiation extending in energy to about 40 electronvolts. Our study indicates the association of the emitted EUV radiation with intraband currents of multi-petahertz frequency, induced in the lowest conduction band of SiO2. To demonstrate the applicability of high-harmonic spectroscopy to solids, we exploit the EUV spectra to gain access to fine details of the energy dispersion profile of the conduction band that are as yet inaccessible by photoemission spectroscopy in wide-bandgap dielectrics. In addition, we use the EUV spectra to trace the attosecond control of the intraband electron motion induced by synthesized optical transients. Our work advances light-wave electronics(5,13-15) in condensed matter into the realm of multi-petahertz frequencies and their attosecond control, and marks the advent of solid-state EUV photonics.