This paper presents a high-performance 5G/6G communication framework tailored explicitly for deep-space missions, integrating multi-level artificial intelligence (AI) to support autonomous and resilient operations beyond Earth. Recognizing unique challenges of space, including extreme latency, bandwidth constraints, and environmental resilience, the proposed framework incorporates advanced network technologies such as network slicing, Multi-access Edge Computing (MEC), and cognitive AI-driven orchestration. Key applications explored include autonomous management of lunar and Martian operations, deep-space robotic missions, and intelligent vehicle control. Through conceptual case studies, the paper illustrates how AI-enhanced local decision-making and adaptive network management significantly improve mission responsiveness, data prioritization, and autonomous operation reliability. Critical challenges such as radiation resilience, cybersecurity, and power efficiency are also addressed, along with potential future directions toward 6G integration, federated learning across space assets, and AI-driven mission design.