Cross-domain Recommendation (CR) has been extensively studied in recent years to alleviate the data sparsity issue in recommender systems by utilizing different domain information. In this work, we focus on the more general Non-overlapping Cross-domain Sequential Recommendation (NCSR) scenario. NCSR is challenging because there are no overlapped entities (e.g., users and items) between domains, and there is only users' implicit feedback and no content information. Previous CR methods cannot solve NCSR well, since (1) they either need extra content to align domains or need explicit domain alignment constraints to reduce the domain discrepancy from domain-invariant features, (2) they pay more attention to users' explicit feedback (i.e., users' rating data) and cannot well capture their sequential interaction patterns, (3) they usually do a single-target cross-domain recommendation task and seldom investigate the dual-target ones. Considering the above challenges, we propose Prompt Learning-based Cross-domain Recommender (PLCR), an automated prompting-based recommendation framework for the NCSR task. Specifically, to address the challenge (1), PLCR resorts to learning domain-invariant and domain-specific representations via its prompt learning component, where the domain alignment constraint is discarded. For challenges (2) and (3), PLCR introduces a pre-trained sequence encoder to learn users' sequential interaction patterns, and conducts a dual-learning target with a separation constraint to enhance recommendations in both domains. Our empirical study on two sub-collections of Amazon demonstrates the advance of PLCR compared with some related SOTA methods.