We have observed that thin films of different undoped non‐conjugated polymers exhibit high conductivity in metal–polymer–metal (M‐P‐M) structures. Below the critical temperature, Tc of the superconductivity of the contacts, a supercurrent in superconductor–polymer–superconductor (S‐P‐S) structures was obtained. Investigations of the current‐voltage characteristics as a function of contact diameter at different locations on the polymer showed that only in some small channels does a high conductivity appear, whereas at other places a reversible on/off switching was obtained. We postulate that the conductivity effect is connected with the phenomenon of electrification. The electrons or holes are supplied by the electrodes which are in contact with the polymer. These injected charge carriers accumulate in localised states on different locations in the polymer matrix. As a result, a large inner electrical field appears in the polymer giving rise to local changes between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). In the high field region the forbidden gap can vanish, resulting in small channels with a high “metallic‐like” conductivity of the polymer. At present, there exists no theory explaining all the peculiarities of the electrical transport observed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)