Mechanically stretchable piezoelectric materials are required by wearable devices, such as self-powering sensors and energy harvesters that can monitor human movement and vital biosignals. Polyvinylidene fluoride (PVDF) is a common choice for making flexible piezoelectric sensors but has limited stretchability (strain to failure). Herein we present a new method of significantly increasing the strain-to-failure of PVDF nanofibre membranes by incorporating amino-functionalised boron nitride nanosheets (BNNSs). The PVDF-BNNS nanocomposites were prepared by firstly dispersing a small amount of BNNSs, less than 1.5 wt%, into a PVDF solution and electrospinning. Compared to the un-modified PVDF, the addition of BNNSs yielded increases of eleven times, twice, and forty times in strength, stretchability, and in-plane thermal conductivity, while piezoelectric performance remains unaffected. The higher thermal conductivity offers significantly better heat dissipation ability of piezoelectric sensors than the un-modified PVDF material.