Recent developments in language models have created new opportunities in air traffic control studies. The current focus is primarily on text and language-based use cases. However, these language models may offer a higher potential impact in the air traffic control domain, thanks to their ability to interact with air traffic environments in an embodied agent form. They also provide a language-like reasoning capability to explain their decisions, which has been a significant roadblock for the implementation of automatic air traffic control. This paper investigates the application of a language model-based agent with function-calling and learning capabilities to resolve air traffic conflicts without human intervention. The main components of this research are foundational large language models, tools that allow the agent to interact with the simulator, and a new concept, the experience library. An innovative part of this research, the experience library, is a vector database that stores synthesized knowledge that agents have learned from interactions with the simulations and language models. To evaluate the performance of our language model-based agent, both open-source and closed-source models were tested. The results of our study reveal significant differences in performance across various configurations of the language model-based agents. The best-performing configuration was able to solve almost all 120 but one imminent conflict scenario, including up to four aircraft at the same time. Most importantly, the agents are able to provide human-level text explanations on traffic situations and conflict resolution strategies.