Let \(K\subset {\mathbb{R}}^ n\) be a compact convex set and let \(E_{\{M_ j\}}(K)\) denote the ultradifferentiable functions of Roumieu type on K. It is shown, that there is no continuous linear extension operator \(T: E_{\{M_ j\}}(K)\to E_{\{M_ j\}}(J)\), if the sequence \((M_ j)\) is regular (\(K\subset\overset\circ J\subset {\mathbb{R}}^ n)\). This especially holds for the Gevrey sequence \(M_ j=(j!)^ s\), \(s>1\). The proof uses the category of tame (F)-spaces and an appropriate variant of the property (DN), which was introduced by D. Vogt to characterize the subspaces of (nuclear) power series spaces of finite type.