Learning general-purpose user representations based on user behavioral logs is an increasingly popular user modeling approach. It benefits from easily available, privacy-friendly yet expressive data, and does not require extensive re-tuning of the upstream user model for different downstream tasks. While this approach has shown promise in search engines and e-commerce applications, its fit for instant messaging platforms, a cornerstone of modern digital communication, remains largely uncharted. We explore this research gap using Snapchat data as a case study. Specifically, we implement a Transformer-based user model with customized training objectives and show that the model can produce high-quality user representations across a broad range of evaluation tasks, among which we introduce three new downstream tasks that concern pivotal topics in user research: user safety, engagement and churn. We also tackle the challenge of efficient extrapolation of long sequences at inference time, by applying a novel positional encoding method.