  We construct hairy black hole solutions in a particular set of
tensor-multi-scalar theories of gravity for which the target-space admits a
Killing vector field with periodic flow that is furthermore the generator of a
one-parameter family of point transformations which leave the whole theory
invariant. As for black holes with scalar hair in general relativity, these
must be rotating objects as the scalarization arises from superradiant
instability. We analyze the most fundamental rotating solutions (winding number
$m=1$) in five theories which differ from each other by the value of the Gauss
curvature of the target-space. The domain of existence of these solutions is
investigated, as well as their hairyness, horizon area and deformation factor.
