Under pressure, metals exhibit increasingly shorter interatomic distances. Intuitively, this response is expected to be accompanied by an increase in the widths of the valence and conduction bands and hence a more pronounced free-electron-like behaviour. But at the densities that can now be achieved experimentally, compression can be so substantial that core electrons overlap. This effect dramatically alters electronic properties from those typically associated with simple free-electron metals such as lithium (Li; refs 1-3) and sodium (Na; refs 4, 5), leading in turn to structurally complex phases and superconductivity with a high critical temperature. But the most intriguing prediction - that the seemingly simple metals Li (ref. 1) and Na (ref. 4) will transform under pressure into insulating states, owing to pairing of alkali atoms - has yet to be experimentally confirmed. Here we report experimental observations of a pressure-induced transformation of Na into an optically transparent phase at ∼200 GPa (corresponding to ∼5.0-fold compression). Experimental and computational data identify the new phase as a wide bandgap dielectric with a six-coordinated, highly distorted double-hexagonal close-packed structure. We attribute the emergence of this dense insulating state not to atom pairing, but to p-d hybridizations of valence electrons and their repulsion by core electrons into the lattice interstices. We expect that such insulating states may also form in other elements and compounds when compression is sufficiently strong that atomic cores start to overlap strongly.