Summary: We show that for every positive integer \(k\) there exists an interval map \(f:I\to I\) such that (1) \(f\) is Li-Yorke chaotic, (2) the inverse limit space \(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum, (3) \(f\) is \(C^{k}\)-smooth, and (4) \(f\) is not \(C^{k+1}\)-smooth. We also show that there exists a \(C^{\infty}\)-smooth \(f\) that satisfies (1) and (2). This answers a recent question of the first author and P. Oprocha from [Proc. Am. Math. Soc. 143, No. 8, 3659–3670 (2015; Zbl 1320.54021)], where the result was proved for \(k=0\). Our study builds on the work of Misiurewicz and Smítal of a family of zero entropy weakly unimodal maps. With the help of a result of Bennett, as well as Blokh’s spectral decomposition theorem, we are also able to show that each \(I_{f}\) contains, for every integer \(i\), a subcontinuum \(C_{i}\) with the following two properties: (i) \(C_{i}\) is \(2^{i}\)-periodic under the shift homeomorphism, and (ii) \(C_{i}\) is a compactification of a topological ray. Finally, we prove that the chaotic attractors we construct are topologically distinct from the one presented by P. Oprocha and the first author.