Trajectory planning for robotic manipulators operating in dynamic orbital debris environments poses significant challenges due to complex obstacle movements and uncertainties. This paper presents Deep Koopman RRT (DK-RRT), an advanced collision-aware motion planning framework integrating deep learning with Koopman operator theory and Rapidly-exploring Random Trees (RRT). DK-RRT leverages deep neural networks to identify efficient nonlinear embeddings of debris dynamics, enhancing Koopman-based predictions and enabling accurate, proactive planning in real-time. By continuously refining predictive models through online sensor feedback, DK-RRT effectively navigates the manipulator through evolving obstacle fields. Simulation studies demonstrate DK-RRT's superior performance in terms of adaptability, robustness, and computational efficiency compared to traditional RRT and conventional Koopman-based planning, highlighting its potential for autonomous space manipulation tasks.