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Artificial intelligence (AI)-driven zero-touch massive network slicing is envisioned to be a disruptive technology in beyond 5G (B5G)/6G, where tenancy would be extended to the final consumer in the form of advanced digital use-cases. In this paper, we propose a novel model-free deep reinforcement learning (DRL) framework, called collaborative statistical Actor-Critic (CS-AC) that enables a scalable and farsighted slice performance management in a 6G-like RAN scenario that is built upon mobile edge computing (MEC) and massive multiple-input multiple-output (mMIMO). In this intent, the proposed CS-AC targets the optimization of the latency cost under a long-term statistical service-level agreement (SLA). In particular, we consider the Q-th delay percentile SLA metric and enforce some slice-specific preset constraints on it. Moreover, to implement distributed learners, we propose a developed variant of soft Actor-Critic (SAC) with less hyperparameter sensitivity. Finally, we present numerical results to showcase the gain of the adopted approach on our built OpenAI-based network slicing environment and verify the performance in terms of latency, SLA Q-th percentile, and time efficiency. To the best of our knowledge, this is the first work that studies the feasibility of an AI-driven approach for massive network slicing under statistical SLA.