The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=\mu_0j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\gamma^\mu\psi(x)$. According to the nomenclature of differential equations, this is a linear, inhomogeneous partial differential equation because the RHS is independent of $A^\mu$. Then, why are interacting field theories called non-linear?