Interest in bioinspired peristaltic pumps has grown in popularity among the scientific community in the last decade thanks to their extreme flexibility and their intrinsic compliance. In this paper, we propose a soft peristaltic pump exploiting ballooning. Our aim is to promote and propel forward the ballooned region by controlling the air pressure between the balloon and an external flexible containment tube, to achieve a peristaltic pumping motion with a simple design and using only one control signal. This paper describes the implementation of the pump and the inlet-pump-outlet system, provides an analytical model to predict the pump performance, and showcases experimental results. We also implement a computer simulation to further characterize the device. We show that it is possible to achieve high volumetric flow rates, up to 4.4 mL/s, with only a single control signal, paving the way for more flexible and easy to manufacture peristaltic pumps.