Exponential functions in prime characteristic

In this note we determine all power series $F(X)\in 1+X\F_p[[X]]$ such that $(F(X+Y))^{-1} F(X)F(Y)$ has only terms of total degree a multiple of $p$. Up to a scalar factor, they are all the series of the form $F(X)=E_p(cX)\cdot G(X^p)$ for some $c\in\F_p$ and $G(X)\in 1+X\F_p[[X]]$, where $E_p(X)=\exp\big(\sum_{i=0}^{\infty}X^{p^i}/p^i\big)$ is the Artin-Hasse exponential.