Energy efficiency, parallel information processing, and unsupervised learning make the human brain a model computing system for unstructured data handling. Different types of oxide memristors can emulate synaptic functions in artificial neuromorphic circuits. However, their cycle‐to‐cycle variability or strict epitaxy requirements remain a challenge for applications in large‐scale neural networks. Here, solution‐processable ferroelectric tunnel junctions (FTJs) with P(VDF‐TrFE) copolymer barriers are reported showing analog memristive behavior with a broad range of accessible conductance states and low energy dissipation of 100 fJ for the onset of depression and 1 pJ for the onset of potentiation by resetting small tunneling currents on nanosecond timescales. Key synaptic functions like programmable synaptic weight, long‐ and short‐term potentiation and depression, paired‐pulse facilitation and depression, and Hebbian and anti‐Hebbian learning through spike shape and timing‐dependent plasticity are demonstrated. In combination with good switching endurance and reproducibility, these results offer a promising outlook on the use of organic FTJ memristors as building blocks in artificial neural networks.