Intracortical brain-computer interfaces (iBCIs) have shown promise for restoring rapid communication to people with neurological disorders such as amyotrophic lateral sclerosis (ALS). However, to maintain high performance over time, iBCIs typically need frequent recalibration to combat changes in the neural recordings that accrue over days. This requires iBCI users to stop using the iBCI and engage in supervised data collection, making the iBCI system hard to use. In this paper, we propose a method that enables self-recalibration of communication iBCIs without interrupting the user. Our method leverages large language models (LMs) to automatically correct errors in iBCI outputs. The self-recalibration process uses these corrected outputs ("pseudo-labels") to continually update the iBCI decoder online. Over a period of more than one year (403 days), we evaluated our Continual Online Recalibration with Pseudo-labels (CORP) framework with one clinical trial participant. CORP achieved a stable decoding accuracy of 93.84% in an online handwriting iBCI task, significantly outperforming other baseline methods. Notably, this is the longest-running iBCI stability demonstration involving a human participant. Our results provide the first evidence for long-term stabilization of a plug-and-play, high-performance communication iBCI, addressing a major barrier for the clinical translation of iBCIs.