Frequently updating Large Language Model (LLM)-based recommender systems to adapt to new user interests -- as done for traditional ones -- is impractical due to high training costs, even with acceleration methods. This work explores adapting to dynamic user interests without any model updates by leveraging In-Context Learning (ICL), which allows LLMs to learn new tasks from few-shot examples provided in the input. Using new-interest examples as the ICL few-shot examples, LLMs may learn real-time interest directly, avoiding the need for model updates. However, existing LLM-based recommenders often lose the in-context learning ability during recommendation tuning, while the original LLM's in-context learning lacks recommendation-specific focus. To address this, we propose RecICL, which customizes recommendation-specific in-context learning for real-time recommendations. RecICL organizes training examples in an in-context learning format, ensuring that in-context learning ability is preserved and aligned with the recommendation task during tuning. Extensive experiments demonstrate RecICL's effectiveness in delivering real-time recommendations without requiring model updates. Our code is available at https://github.com/ym689/rec_icl.