As spacecraft journey further from Earth with more complex missions, systems of greater autonomy and onboard intelligence are called for. Reducing reliance on human-based mission control becomes increasingly critical if we are to increase our rate of solar-system-wide exploration. Recent work has explored AI-based goal-oriented systems to increase the level of autonomy in mission execution. These systems make use of symbolic reasoning managers to make inferences from the state of a spacecraft and a handcrafted knowledge base, enabling autonomous generation of tasks and re-planning. Such systems have proven to be successful in controlled cases, but they are difficult to implement as they require human-crafted ontological models to allow the spacecraft to understand the world. Reinforcement learning has been applied to train robotic agents to pursue a goal. A new architecture for autonomy is called for. This work explores the application of Large Language Models (LLMs) as the high-level control system of a spacecraft. Using a systems engineering approach, this work presents the design and development of an agentic spacecraft controller by leveraging an LLM as a reasoning engine, to evaluate the utility of such an architecture in achieving higher levels of spacecraft autonomy. A series of deep space mission scenarios simulated within the popular game engine Kerbal Space Program (KSP) are used as case studies to evaluate the implementation against the requirements. It is shown the reasoning and planning abilities of present-day LLMs do not scale well as the complexity of a mission increases, but this can be alleviated with adequate prompting frameworks and strategic selection of the agent's level of authority over the host spacecraft. This research evaluates the potential of LLMs in augmenting autonomous decision-making systems for future robotic space applications.