  A new family of non-Hermitian PT-symmetric quantum models is proposed in
which the Hamiltonians $H=T+V$ are finite-dimensional and in which the
dynamical-input potential $V$ is multi-parametric and non-local. The choice is
supported by the exact solvability of Schr\"{o}dinger equation and by the well
known fact that in PT-symmetric models a non-locality is already present due to
the generic kinematical non-diagonality of the Hermitizing metrics $\Theta$.
For a subfamily of our $H$s, also {\em all\,} of the eligible metrics $\Theta$
appear obtainable in closed form.
