Authorship attribution research has traditionally focused on long-form, expressive texts; however, interactions with large language models (LLMs) are typically brief and task-driven prompts. This raises a fundamental question: do such prompts contain a stable, author-identifiable, and distinctive signal? We introduce PromptPrint, a systematic study of prompt-based identity, the hypothesis that a user's habitual vocabulary, syntax, and discourse patterns form a learnable behavioral biometric. Using 20,680 real prompts from 1,034 users, we establish three key findings. First, lexical representations significantly outperform semantic encoders, supporting the "lexical stability hypothesis": identity is primarily encoded in surface-level word choice rather than abstract intent. Second, stylometric features exhibit a "uniqueness-consistency paradox": users are highly distinctive across the population, yet behaviorally inconsistent across contexts. Third, adversarial analysis reveals a clear vulnerability spectrum: identity signals are robust to minor lexical perturbations but degrade substantially under semantic paraphrasing. Overall, our results demonstrate strong identification performance at scale, establishing prompt-based identity as a viable behavioral biometric. This work introduces a new perspective on user modeling in LLM interactions, with important implications for security and privacy. Data and code will be released upon the acceptance of our work.