Recent trends are emerging in the use of Large Language Models (LLMs) as autonomous agents that take actions based on the content of the user text prompts. We intend to apply these concepts to the field of Control in space, enabling LLMs to play a significant role in the decision-making process for autonomous satellite operations. As a first step towards this goal, we have developed a pure LLM-based solution for the Kerbal Space Program Differential Games (KSPDG) challenge, a public software design competition where participants create autonomous agents for maneuvering satellites involved in non-cooperative space operations, running on the KSP game engine. Our approach leverages prompt engineering, few-shot prompting, and fine-tuning techniques to create an effective LLM-based agent that ranked 2nd in the competition. To the best of our knowledge, this work pioneers the integration of LLM agents into space research. The project comprises several open repositories to facilitate replication and further research. The codebase is accessible on \href{https://github.com/ARCLab-MIT/kspdg}{GitHub}, while the trained models and datasets are available on \href{https://huggingface.co/OhhTuRnz}{Hugging Face}. Additionally, experiment tracking and detailed results can be reviewed on \href{https://wandb.ai/carrusk/huggingface}{Weights \& Biases