Playing partial information, restricted communication, cooperative (PIRCC) games with humans have proven challenging for AI, due to our reliance on social dynamics and sophisticated cognitive techniques. Yet, recent advances in generative AI may change this situation through new forms of human-AI collaboration. This paper investigates how teams of players interact with an AI assistant in the PIRCC game Codenames and the impact this has on cognition, social dynamics, and player experience. We observed gameplay and conducted post-game focus groups with 54 participants across ten groups. Each group played three rounds of Codenames, with an AI assistant supporting Cluegivers. We found the AI assistant enhanced players' convergent and divergent thinking, but interfered with formation of team mental models, highlighting a tension in the use of AI in creative team scenarios. The presence of the AI challenged many players' understanding of the 'spirit of the game'. Furthermore, the presence of the AI assistants weakened social connections between human teammates, but strengthened connections across teams. This paper provides an empirical account of an AI assistant's effect on cognition, social dynamics, and player experience in Codenames. We highlight the opportunities and challenges that arise when designing hybrid digital boardgames that include AI assistants.