  The possibility that dark matter may be in the form of a Bose-Einstein
Condensate (BEC) has been extensively explored at galactic scale. In
particular, good fits for the galactic rotations curves have been obtained, and
upper limits for the dark matter particle mass and scattering length have been
estimated. In the present paper we extend the investigation of the properties
of the BEC dark matter to the galactic cluster scale, involving dark matter
dominated astrophysical systems formed of thousands of galaxies each. By
considering that one of the major components of a galactic cluster, the
intra-cluster hot gas, is described by King's $\beta$-model, and that both
intra-cluster gas and dark matter are in hydrostatic equilibrium, bound by the
same total mass profile, we derive the mass and density profiles of the BEC
dark matter. In our analysis we consider several theoretical models,
corresponding to isothermal hot gas and zero temperature BEC dark matter,
non-isothermal gas and zero temperature dark matter, and isothermal gas and
finite temperature BEC, respectively. The properties of the finite temperature
BEC dark matter cluster are investigated in detail numerically. We compare our
theoretical results with the observational data of 106 galactic clusters. Using
a least-squares fitting, as well as the observational results for the dark
matter self-interaction cross section, we obtain some upper bounds for the mass
and scattering length of the dark matter particle. Our results suggest that the
mass of the dark matter particle is of the order of $\mu $eV, while the
scattering length has values in the range of $10^{-7}$ fm.
