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Over the past few decades, the measurement precision of some pulsar-timing experiments has advanced from ~10 us to ~10 ns, revealing many subtle phenomena. Such high precision demands both careful data handling and sophisticated timing models to avoid systematic error. To achieve these goals, we present PINT (PINT Is Not Tempo3), a high-precision Python pulsar timing data analysis package, which is hosted on GitHub and available on Python Package Index (PyPI) as pint-pulsar. PINT is well-tested, validated, object-oriented, and modular, enabling interactive data analysis and providing an extensible and flexible development platform for timing applications. It utilizes well-debugged public Python packages (e.g., the NumPy and Astropy libraries) and modern software development schemes (e.g., version control and efficient development with git and GitHub) and a continually expanding test suite for improved reliability, accuracy, and reproducibility. PINT is developed and implemented without referring to, copying, or transcribing the code from other traditional pulsar timing software packages (e.g., TEMPO and TEMPO2) and therefore provides a robust tool for cross-checking timing analyses and simulating pulse arrival times. In this paper, we describe the design, usage, and validation of PINT, and we compare timing results between it and TEMPO and TEMPO2.