Broadband dielectric spectroscopy is employed to investigate the molecular dynamics of cis-1,4-polyisoprene (PI) in thin layers down to thicknesses comparable with the chain extension. The segmental and the normal mode dynamics of the PI chains is found not to depend on the layer thickness. Additionally, a novel confinement-induced relaxation process is detected. It is assigned to fluctuations of terminal subchains which are formed due to the immobilization of chain segments at an interface. Most of the experimental findings can be well described by simulations of the chains as ideal random walks.