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Massive grant-free transmission and cell-free wireless communication systems have emerged as pivotal enablers for massive machine-type communication. This paper proposes a deep-unfolding-based joint activity and data detection (DU-JAD) algorithm for massive grant-free transmission in cell-free systems. We first formulate a joint activity and data detection optimization problem, which we solve approximately using forward-backward splitting (FBS). We then apply deep unfolding to FBS to optimize algorithm parameters using machine learning. In order to improve data detection (DD) performance, reduce algorithm complexity, and enhance active user detection (AUD), we employ a momentum strategy, an approximate posterior mean estimator, and a novel soft-output AUD module, respectively. Simulation results confirm the efficacy of DU-JAD for AUD and DD.