  The formation of the first stars in the high-redshift Universe is a sensitive
probe of the small-scale, particle physics nature of dark matter (DM). We carry
out cosmological simulations of primordial star formation in ultra-light,
axion-like particle DM cosmology, with masses of $10^{-22}$ and $10^{-21}\,{\rm
eV}$, with de Broglie wavelengths approaching galactic scales ($\sim$kpc). The
onset of star formation is delayed, and shifted to more massive host
structures. For the lightest DM particle mass explored here, first stars form
at $z \sim 7$ in structures with $\sim 10^9\,{\rm M}_\odot$, compared to the
standard minihalo environment within the $\Lambda$ cold dark matter
($\Lambda$CDM) cosmology, where $z \sim 20 - 30$ and $\sim 10^5 - 10^6\,{\rm
M}_\odot$. Despite this greatly altered DM host environment, the thermodynamic
behaviour of the metal-free gas as it collapses into the DM potential well
asymptotically approaches a very similar evolutionary track. Thus, the
fragmentation properties are predicted to remain the same as in $\Lambda$CDM
cosmology, implying a similar mass scale for the first stars. These results
predict intense starbursts in the axion cosmologies, which may be amenable to
observations with the {\it James Webb Space Telescope}.
