The paper presents a novel computational framework, the Deep xVA Solver, for portfolio-wide counterparty credit risk management. It addresses the challenge of high-dimensional risk factors by reformulating xVA computations as a system of backward stochastic differential equations (BSDEs) and solving them using deep neural networks. The method is built on a rigorous financial model that incorporates default risk, funding costs, and collateral agreements, leading to a pre-default xVA BSDE under the reference filtration. The algorithm parametrizes the hedging strategy at each time step by neural networks and optimizes them via stochastic gradient descent, making it feasible for high-dimensional problems (tested up to 100 assets). Key contributions include the ability to compute both non-recursive xVAs (CVA, DVA) and recursive xVAs (FVA) through a recursive application of the solver, and the simultaneous generation of hedge ratios and dynamic risk measures. Numerical experiments demonstrate basis point accuracy, efficient collateral simulation, and robustness across various test cases. The framework is adaptable to different xVA specifications and offers a significant improvement over traditional nested Monte Carlo and PDE methods in high-dimensional settings.