  We assume dark matter to be a cosmological self-gravitating Bose-Einstein
condensate of non-relativistic ultralight scalar particles with competing
gravitational and repulsive contact interactions and investigate the
observational implications of such model. The system is unstable to the
formation of stationary self-bound structures that minimize the energy
functional. These cosmological superfluid droplets, which are the smallest
possible gravitationally bound dark matter structures, exhibit a universal mass
profile and a corresponding universal rotation curve. Assuming a hierarchical
structure formation scenario where granular dark matter haloes grow around
these primordial stationary droplets, the model predicts cored haloes with
rotation curves that obey a single universal equation in the inner region ($r
\lesssim 1$ kpc). A simultaneous fit to a selection of galaxies from the SPARC
database chosen with the sole criterion of being strongly dark matter dominated
even within the innermost region, indicates that the observational data are
consistent with the presence of a Bose-Einstein condensate of ultralight scalar
particles of mass $m \simeq 2.2 \times 10^{-22}$ eV c$^{-2}$ and repulsive
self-interactions characterized by a scattering length $a_s \simeq 7.8 \times
10^{-77}$ m. Such small self-interactions have profound consequences on
cosmological scales. They induce a natural minimum scale length for the size of
dark matter structures that makes all cores similar in length ($\sim 1$ kpc)
and contributes to lower their central densities.
