This paper describes carrier transport mechanisms in polyethylene terephthalate (PET) films and porous PET-based membranes (PMs) obtained by irradiating pristine PET film with swift heavy ions, with subsequent chemical etching in an alkali (NaOH) solution. The obtained PMs had through nanochannels (pores) with an average diameter of 720-750 nm. We observed that in the temperature range 240-300 K, the current-voltage characteristics I(V) of the initial Cu|PET|Cu structure obeyed the improved Mott--Gurney law, which is based on the Mark--Helfrich model for a space-charge-limited current (SCLC) mechanism for electron transport. It was found for the first time that creation of nanochannels in PMs resulted in a significant increase in the electric current density (by about three orders of magnitude) while maintaining the SCLC mechanism. The enhanced current density is explained by the formation of a highly conductive layer along the inner surface of the walls of the nanochannel that are covered with carboxyl end groups, which are created by alkaline hydrolysis. According to the model, the surface states formed by these groups enable the drift of additional electrons injected from the copper electrodes under the action of the bias voltage. Keywords: polyethylene terephthalate, electron transport, space-charge-limited current mechanism, Mark--Helfrich injection model.