Multifunctional capability, flexible design, rugged lightweight construction and self-powered operation are desired attributes for electronics that directly interface with the human body or with advanced robotic systems. For these applications, piezoelectric materials, in forms that offer the ability to bend and stretch, are attractive for pressure/force sensors and mechanical energy harvesters. Here, we introduce a large area, flexible piezoelectric material that consists of sheets of electrospun fibres of the polymer poly[(vinylidenefluoride-co-trifluoroethylene]. The flow and mechanical conditions associated with the spinning process yield free-standing, three-dimensional architectures of aligned arrangements of such fibres, in which the polymer chains adopt strongly preferential orientations. The resulting material offers exceptional piezoelectric characteristics, to enable ultra-high sensitivity for measuring pressure, even at exceptionally small values (0.1 Pa). Quantitative analysis provides detailed insights into the pressure sensing mechanisms, and establishes engineering design rules. Potential applications range from self-powered micro-mechanical elements, to self-balancing robots and sensitive impact detectors.