Recent advances in artificial intelligence have enhanced the abilities of mobile robots in dealing with complex and dynamic scenarios. However, to enable computationally intensive algorithms to be executed locally in multitask robots with low latency and high efficiency, innovations in computing hardware are required. Here, we report TianjicX, a neuromorphic computing hardware that can support true concurrent execution of multiple cross-computing-paradigm neural network (NN) models with various coordination manners for robotics. With spatiotemporal elasticity, TianjicX can support adaptive allocation of computing resources and scheduling of execution time for each task. Key to this approach is a high-level model, "Rivulet," which bridges the gap between robotic-level requirements and hardware implementations. It abstracts the execution of NN tasks through distribution of static data and streaming of dynamic data to form the basic activity context, adopts time and space slices to achieve elastic resource allocation for each activity, and performs configurable hybrid synchronous-asynchronous grouping. Thereby, Rivulet is capable of supporting independent and interactive execution. Building on Rivulet with hardware design for realizing spatiotemporal elasticity, a 28-nanometer TianjicX neuromorphic chip with event-driven, high parallelism, low latency, and low power was developed. Using a single TianjicX chip and a specially developed compiler stack, we built a multi-intelligent-tasking mobile robot, Tianjicat, to perform a cat-and-mouse game. Multiple tasks, including sound recognition and tracking, object recognition, obstacle avoidance, and decision-making, can be concurrently executed. Compared with NVIDIA Jetson TX2, latency is substantially reduced by 79.09 times, and dynamic power is reduced by 50.66%.