Let \(N\) be a finite abelian extension of the function field \(K= \mathbb F_q (T)\) with Galois group \(\Gamma\) and \(O_N\) the integral closure of \(O_K= \mathbb F_q [T]\) in \(N\). Supposing that no prime ideal of \(O_N\) is wildly ramified in \(N\), \textit{R. J. Chapman} [J. Lond. Math. Soc., II. Ser. 44, 250-260 (1991; Zbl 0749.11049)] proved that \(O_N \simeq A(N/K) = O_K \Gamma\), where \(A(N/K) = \{ x \in K\Gamma \mid x O_N \subset O_N \}\) denotes the associated order. As a counterpart to this result, the present paper shows that \(O_N \not \simeq A(N/K)\) if \(N\) is obtained from \(K\) by adjoining the torsion points of the Carlitz module arising from powers \(P^n, n \geq 2\) (\(n \geq 3\) if \(q=2\)), of a linear polynomial \(P \in O_K\). In this situation, \(N/K\) is wildly ramified at \(P\).