  In hierarchical models of structure formation, the first galaxies form in
low-mass dark matter potential wells, probing the behavior of dark matter on
kiloparsec (kpc) scales. Even though these objects are below the detection
threshold of current telescopes, future missions will open an observational
window into this emergent world. In this Letter we investigate how the first
galaxies are assembled in a `fuzzy' dark matter (FDM) cosmology where dark
matter is an ultralight $\sim 10^{-22}$~eV boson and the primordial stars are
expected to form along dense dark matter filaments. Using a first-of-its-kind
cosmological hydrodynamical simulation, we explore the interplay between
baryonic physics and unique wavelike features inherent to FDM. In our
simulation, the dark matter filaments show coherent interference patterns on
the boson de Broglie scale and develop cylindrical soliton-like cores which are
unstable under gravity and collapse into kpc-scale spherical solitons. Features
of the dark matter distribution are largely unaffected by the baryonic
feedback. On the contrary, the distributions of gas and stars, which do form
along the entire filament, exhibit central cores imprinted by dark matter -- a
smoking gun signature of FDM.
