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A new family of non-Hermitian PT-symmetric quantum models is proposed inwhich the Hamiltonians $H=T+V$ are finite-dimensional and in which thedynamical-input potential $V$ is multi-parametric and non-local. The choice issupported by the exact solvability of Schr\"{o}dinger equation and by the wellknown fact that in PT-symmetric models a non-locality is already present due tothe 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.

PT-symmetric model with an interplay between kinematical and dynamical non-localities