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We use magnetohydrodynamic levitation as a means to create a soft, elastomeric, solenoid-driven pump (ESP). We present a theoretical framework and fabrication of a pump designed to address the unique challenges of soft robotics, maintaining pumping performance under deformation. Using a permanent magnet as a piston and ferrofluid as a liquid seal, we model and construct a deformable displacement pump. The magnet is driven back and forth along the length of a flexible core tube by a series of solenoids made of thin conductive wire. The magnet piston is kept concentric within the tube by Maxwell stresses within the ferrofluid and magnetohydrodynamic levitation, as viscous lift pressure is created due to its forward velocity. The centering of the magnet reduces shear stresses during pumping and improves efficiency. We provide a predictive model and capture the transient nonlinear dynamics of the magnet during operation, leading to a parametric performance curve characterizing the ESP, enabling goal-driven design. In our experimental validation, we report a shut-off pressure of 2 to 8 kPa and run-out flow rate of 50 to 320 mL⋅min −1 , while subject to deformation of its own length scale, drawing a total of 0.17 W. This performance leads to the highest reported duty point (i.e., pressure and flow rate provided under load) for a pump that operates under deformation of its own length scale. We then integrate the pump into an elastomeric chassis and squeeze it through a tortuous pathway while providing continuous fluid pressure and flow rate; the vehicle then emerges at the other end and propels itself swimming.