  The conventional cold, particle interpretation of dark matter (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 ($\psi$DM) composed of a non-relativistic Bose-Einstein condensate,
so the uncertainty principle counters gravity below a Jeans scale. Here we
achieve the first cosmological simulations of this quantum state at
unprecedentedly high resolution capable of resolving dwarf galaxies, with only
one free parameter, $\bf{m_B}$, the boson mass. We demonstrate the large scale
structure of this $\psi$DM simulation is indistinguishable from CDM, as
desired, but differs radically inside galaxies. Connected filaments and
collapsed haloes form a large interference network, with gravitationally
self-bound solitonic cores inside every galaxy surrounded by extended haloes of
fluctuating density granules. These results allow us to determine
$\bf{m_B=(8.1^{+1.6}_{-1.7})\times 10^{-23}~eV}$ 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. Suppression of small structures means the
onset of galaxy formation for $\psi$DM is substantially delayed relative to
CDM, appearing at $\bf{z\lesssim 13}$ in our simulations.
