Massive MIMO is considered a key enabler to support massive machine-type communication (mMTC). While massive access schemes have been extensively analyzed for co-located massive MIMO arrays, this paper explores activity detection in grant-free random access for mMTC within the context of cell-free massive MIMO systems, employing distributed antenna arrays. This sparse support recovery of device activity status is performed by a finite cluster of access points (APs) from a large number of geographically distributed APs collaborating to serve a larger number of devices. Active devices transmit non-orthogonal pilot sequences to APs, which forward the received signals to a central processing unit (CPU) for collaborative activity detection. This paper proposes a simple and efficient data-driven algorithm tailored for device activity detection, implemented centrally at the CPU. Furthermore, the study assesses the algorithm's robustness to input perturbations and examines the effects of adopting fixed-point representation on its performance.