Using fully-kinetic plasma simulations, we study the non-resonant (Bell) streaming instability driven by energetic leptons. We identify the necessary conditions to drive it and the differences from the standard proton-driven case in both linear and saturated stages. A simple analytic theory is presented to explain simulations. Our findings are crucial for understanding the phenomenology of astrophysical environments where only electrons may be accelerated (e.g., oblique shocks) or where relativistic pairs are produced (e.g., around pulsar wind nebulae).