We report a memory resistance (memristor) behavior with nonlinear current-voltage characteristics and bipolar hysteretic resistance switching in the nanocolumnar manganite (LSMO) films. The switching from a high (HRS) to a low (LRS) resistance occurs at a bias field ~1 MV/cm. Applied electric field drops mostly at the insulating interfacial LSMO layer and couples to correlated polarons at the LSMO(111)/LSMO(111) vertical interfaces. The observed memristance behaviour has an electronic (polaronic) origin and is caused by an electric-field-controlled Jahn-Teller (JT) effect, followed by the orbital reconstruction and formation of a metastable orbitally disordered interfacial phase (LRS). Compared to the earlier reported ionic memristor in Ti-O films, an electronic (polaronic) nano-sized LSMO memristor shows an additional (re-entrant) LRS-HRS switching at higher fields because of the second minimum in the elastic energy of a JT system.