Organic electronics is emerging for large-area applications such as photovoltaic cells, rollable displays or electronic paper. Its future development and integration will require a simple, low-power organic memory, that can be written, erased and readout electrically. Here we demonstrate a non-volatile memory in which the ferroelectric polarisation state of an organic tunnel barrier encodes the stored information and sets the readout tunnel current. We use high-sensitivity piezoresponse force microscopy to show that films as thin as one or two layers of ferroelectric poly(vinylidene fluoride) remain switchable with low voltages. Submicron junctions based on these films display tunnel electroresistance reaching 1,000% at room temperature that is driven by ferroelectric switching and explained by electrostatic effects in a direct tunnelling regime. Our findings provide a path to develop low-cost, large-scale arrays of organic ferroelectric tunnel junctions on silicon or flexible substrates.