  We construct a family of asymptotically flat, rotating black holes with
scalar hair and a regular horizon, within five dimensional Einstein's gravity
minimally coupled to a complex, massive scalar field doublet. These solutions
are supported by rotation and have no static limit. They are described by their
mass $M$, two equal angular momenta $J_1=J_2\equiv J$ and a conserved Noether
charge $Q$, measuring the scalar hair. For vanishing horizon size the solutions
reduce to five dimensional boson stars. In the limit of vanishing Noether
charge density, the scalar field becomes point-wise arbitrarily small and the
geometry becomes, locally, arbitrarily close to that of a specific set of
Myers-Perry black holes (MPBHs); but there remains a global difference with
respect to the latter, manifest in a finite mass gap. Thus, the scalar hair
never becomes a linear perturbation of MPBHs. This is a qualitative difference
when compared to Kerr black holes with scalar hair~\cite{Herdeiro:2014goa}.
Whereas the existence of the latter can be anticipated in linear theory, from
the existence of scalar bound states on the Kerr geometry (i.e. scalar clouds),
the hair of these MPBHs is intrinsically non-linear.
