Let \(\mathbb F\) be a finite field of \(q\) elements. Let \(\mathbb F((X^{-1}))\) be the field of formal Laurent series \[ \left\{\sum_{n=n_0}^\infty a_n X^{-n}\mid a_n\in\mathbb F, n_0\in\mathbb Z\right\}. \] Let \(\beta\in\mathbb F((X^{-1}))\) with \(\|\beta\|>1\). The \(\beta\)-expansion of \(x\in\mathbb F((X^{-1}))\), \(\|x\|<1\), is \(x=\sum_{n=1}^\infty \frac{\varepsilon_n(x)}{\beta^n}\). Denote \(\omega_n(x)=\sum_{j=1}^n \frac{\varepsilon_j(x)}{\beta^j}\). The following theorems are proved. Almost every \(x\) is approximated by its convergents with order \(-n\deg\beta\), in other words \[ \lim_{n\to\infty} \frac{1}{n} \log_q \|x-\omega_n(x)\|=-\deg\beta. \] For an increasing function \(\phi:\mathbb N\to\mathbb R^+\) with \(\phi(n)\to\infty\) as \(n\to\infty\) define \(\eta=\liminf_{n\to\infty}\frac{\phi(n)}{n}\) and \[ A_\phi=\bigl\{x\in\mathbb F((X^{-1})), \|x\|<1 \bigm| \liminf_{n\to\infty}\frac{1}{\phi(n)} \log_q \|x-\omega_n(x)\|=-1\bigr\}. \] If \(\eta\in[\deg\beta,\infty)\) then the Hausdorff dimension \(\dim A_\phi=\frac{\deg\beta}{\eta}\). If \(\eta\in[0,\deg\beta)\) or \(\eta=\infty\) then \(\dim A_\phi=0\).