The aim of this paper is to characterize the smallest $2$-Pisot number of degree $n \geq 2$ in the case of Laurent power series $\mathbb{F}_{q}((X^{-1}))$ with $q$ distinct from the power of $2$ over a finite field $\mathbb{F}_{q}$ by giving explicitly its minimal polynomial. Indeed, we prove that its minimal polynomial is given by $\Lambda_{n}(Y) = Y^{n} - \alpha X(X+1) Y^{n-1} - \alpha^2 X^3 Y^{n-2} + \alpha^{n}$. We show in particular that the sequence of smallest $2$-pisot numbers of degree $n$ is decreasing and converges to $(\alpha X^2, \alpha X)$ where we suppose that $\alpha$ is the least element of $\mathbb{F}_{q} \setminus \{0\}$.