Mobile robots are increasingly deployed in cluttered environments with movable objects, posing challenges for traditional methods that prohibit interaction. In such settings, the mobile robot must go beyond traditional obstacle avoidance, leveraging pushing or nudging strategies to accomplish its goals. While research in pushing-based robotics is growing, evaluations rely on ad hoc setups, limiting reproducibility and cross-comparison. To address this, we present Bench-Push, the first unified benchmark for pushing-based mobile robot navigation and manipulation tasks. Bench-Push includes multiple components: 1) a comprehensive range of simulated environments that capture the fundamental challenges in pushing-based tasks, including navigating a maze with movable obstacles, autonomous ship navigation in ice-covered waters, box delivery, and area clearing, each with varying levels of complexity; 2) novel evaluation metrics to capture efficiency, interaction effort, and partial task completion; and 3) demonstrations using Bench-Push to evaluate example implementations of established baselines across environments. Bench-Push is open-sourced as a Python library with a modular design. The code, documentation, and trained models can be found at https://github.com/IvanIZ/BenchNPIN.