Mobile manipulation is a critical capability for robots operating in diverse, real-world environments. However, manipulating deformable objects and materials remains a major challenge for existing robot learning algorithms. While various benchmarks have been proposed to evaluate manipulation strategies with rigid objects, there is still a notable lack of standardized benchmarks that address mobile manipulation tasks involving deformable objects. To address this gap, we introduce MoDeSuite, the first Mobile Manipulation Deformable Object task suite, designed specifically for robot learning. MoDeSuite consists of eight distinct mobile manipulation tasks covering both elastic objects and deformable objects, each presenting a unique challenge inspired by real-world robot applications. Success in these tasks requires effective collaboration between the robot's base and manipulator, as well as the ability to exploit the deformability of the objects. To evaluate and demonstrate the use of the proposed benchmark, we train two state-of-the-art reinforcement learning algorithms and two imitation learning algorithms, highlighting the difficulties encountered and showing their performance in simulation. Furthermore, we demonstrate the practical relevance of the suite by deploying the trained policies directly into the real world with the Spot robot, showcasing the potential for sim-to-real transfer. We expect that MoDeSuite will open a novel research domain in mobile manipulation involving deformable objects. Find more details, code, and videos at https://sites.google.com/view/modesuite/home.