In a Hilbert space $H$, based on inertial dynamics with dry friction damping, we introduce a new class of proximal-gradient algorithms with finite convergence properties. The function $f : H → R$ to minimize is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient. The dry friction damping function $φ : H → R^+$ is convex with a sharp minimum at the origin, (typically $φ(x) = r x$ with $r > 0$). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. This algorithm naturally occurs as a discrete temporal version of an inertial differential inclusion involving viscous and dry friction together. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero. Then, replacing the potential function $f$ by its Moreau envelope, we extend the results to the case of a nonsmooth convex function $f$. In this case, the algorithm involves the proximal operators of $f$ and $φ$ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to new splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of two variants of the Nesterov accelerated method with dry friction.