  In this paper, we consider the function field analogue of the Lehmer's
totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $\varphi(q,p(x))$ be the
Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$
is a finite field with $q$ elements. We prove that $\varphi(q,p(x))|(q^{{\rm
deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \;
p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or
(iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all
irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one
each of degree $1, 2$ and $3$.
