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Herein, point light detection and ranging inertial odometry (LIO) is presented: a robust and high‐bandwidth light detection and ranging (LiDAR) inertial odometry with the capability to estimate extremely aggressive robotic motions. Point‐LIO has two key novelties. The first one is a point‐by‐point LIO framework that updates the state at each LiDAR point measurement. This framework allows an extremely high‐frequency odometry output, significantly increases the odometry bandwidth, and fundamentally removes the artificial in‐frame motion distortion. The second one is a stochastic process‐augmented kinematic model which models the IMU measurement as an output. This new modeling method enables accurate localization and reliable mapping for aggressive motions even with inertial measurement unit (IMU) measurements saturated in the middle of the motion. Various real‐world experiments are conducted for performance evaluation. Overall, Point‐LIO is capable to provide accurate, high‐frequency odometry (4–8 kHz) and reliable mapping under severe vibrations and aggressive motions with high angular velocity (75 rad s−1) beyond the IMU measuring ranges. Furthermore, an exhaustive benchmark comparison is conducted. Point‐LIO achieves consistently comparable accuracy and time consumption. Finally, two example applications of Point‐LIO are demonstrated, one is a racing drone and the other is a self‐rotating unmanned aerial vehicle, both have aggressive motions.