Bosonic ultra-light dark matter (ULDM) would form cored density distributions at the centre of galaxies. Numerical simulations suggest a scaling relation between the core and the virialised host halo. This prediction lends itself to observational tests, because it implies that the peak circular velocity, measured for the host halo in the outskirts of the galaxy, should approximately repeat itself in the central region. Contrasting this prediction to the measured rotation curves of dozens of well-resolved galaxies, we show that ULDM in the mass range $m\sim (10^{-22}\div 10^{-21})$~eV, which has been invoked as a possible solution to the small-scale puzzles of $\Lambda$CDM, is in tension with the data. Analysis of the Milky Way inner gravitational potential could probe ULDM up to $m\lesssim 10^{-19}$~eV.