  We present the first rigorous study of nonlinear wave equations on extremal
black hole spacetimes without any symmetry assumptions on the solution.
Specifically, we prove global existence with asymptotic blow-up for solutions
to nonlinear wave equations satisfying the null condition on extremal
Reissner-Nordstr\"{o}m backgrounds. This result shows that the extremal horizon
instability persists in model nonlinear theories. Our proof crucially relies on
a new vector field method that allows us to obtain almost sharp decay
estimates.
