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The conventional cold-particle interpretation of dark matter (known as 'cold dark matter', or CDM) still lacks laboratory support and struggles with the basic properties of common dwarf galaxies, which have surprisingly uniform central masses and shallow density profiles. In contrast, galaxies predicted by CDM extend to much lower masses, with steeper, singular profiles. This tension motivates cold, wavelike dark matter (ψDM) composed of a non-relativistic Bose-Einstein condensate, so the uncertainty principle counters gravity below a Jeans scale. Here we achieve cosmological simulations of this quantum state at unprecedentedly high resolution capable of resolving dwarf galaxies, with only one free parameter, mB, the boson mass. We demonstrate the large-scale structure is indistinguishable from CDM, as desired, but differs radically inside galaxies where quantum interference forms solitonic cores surrounded by extended haloes of fluctuating density granules. These results allow us to determine eV using stellar phase-space distributions in dwarf spheroidal galaxies. Denser, more massive solitons are predicted for Milky Way sized galaxies, providing a substantial seed to help explain early spheroid formation. The onset of galaxy formation is substantially delayed relative to CDM, appearing at redshift z ≲ 13 in our simulations. [ABSTRACT FROM AUTHOR]