  We study the structure of galactic halos within a scalar dark matter model,
endowed with a repulsive quartic self-interaction, capable of undergoing the
superfluid phase transition in high-density regions. We demonstrate that the
thermalized cores are prone to fragmentation into superfluid droplets due to
the Jeans instability. Furthermore, since cores of astrophysical size may be
generated only when most of the particles comprising the halo reside in a
highly degenerate phase-space, the well-known bound on the dark matter
self-interaction cross section inferred from the collision of clusters needs to
be revised, accounting for the enhancement of the interaction rate due to
degeneracy. As a result, generation of kpc-size superfluid solitons, within the
parameter subspace consistent with the Bullet Cluster bound, requires dark
matter particles to be ultra-light.
