We analyze the convergence of a continuous interior penalty (CIP) method for a singularly perturbed fourth-order elliptic problem on a layer-adapted mesh. On this anisotropic mesh, we prove under reasonable assumptions uniform convergence of almost order k-1 for finite elements of degree k2. This result is of better order than the known robust result on standard meshes. A by-product of our analysis is an analytic lower bound for the penalty of the symmetric CIP method. Finally, our convergence result is verified numerically. (c) 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 838-861, 2014