Non-terrestrial networks (NTNs) are essential for ubiquitous connectivity, providing coverage in remote and underserved areas. However, since NTNs are currently operated independently, they face challenges such as isolation, limited scalability, and high operational costs. Integrating satellite constellations with terrestrial networks offers a way to address these limitations while enabling adaptive and cost-efficient connectivity through the application of Artificial Intelligence (AI) models. This paper introduces Space-O-RAN, a framework that extends Open Radio Access Network (RAN) principles to NTNs. It employs hierarchical closed-loop control with distributed Space RAN Intelligent Controllers (Space-RICs) to dynamically manage and optimize operations across both domains. To enable adaptive resource allocation and network orchestration, the proposed architecture integrates real-time satellite optimization and control with AI-driven management and digital twin (DT) modeling. It incorporates distributed Space Applications (sApps) and dApps to ensure robust performance in in highly dynamic orbital environments. A core feature is dynamic link-interface mapping, which allows network functions to adapt to specific application requirements and changing link conditions using all physical links on the satellite. Simulation results evaluate its feasibility by analyzing latency constraints across different NTN link types, demonstrating that intra-cluster coordination operates within viable signaling delay bounds, while offloading non-real-time tasks to ground infrastructure enhances scalability toward sixth-generation (6G) networks.