  We derive a Hamiltonian structure for the $N$-particle hyperbolic spin
Ruijsenaars-Schneider model by means of Poisson reduction of a suitable initial
phase space. This phase space is realised as the direct product of the
Heisenberg double of a factorisable Lie group with another symplectic manifold
that is a certain deformation of the standard canonical relations for $N\ell$
conjugate pairs of dynamical variables. We show that the model enjoys the
Poisson-Lie symmetry of the spin group ${\rm GL}_{\ell}({\mathbb C})$ which
explains its superintegrability. Our results are obtained in the formalism of
the classical $r$-matrix and they are compatible with the recent findings on
the different Hamiltonian structure of the model established in the framework
of the quasi-Hamiltonian reduction applied to a quasi-Poisson manifold.
