Epitaxy of perovskite oxide ferroelectric heterostructures with large lattice misfit is crucial for numerous emerging applications. Here we demonstrate cube-on-cube-type epitaxial growth of BaTiO3 films on strongly mismatched (001)SrTiO3 single-crystal substrates for the films with thicknesses significantly larger than that of misfit relaxation. The films experience strain originating from the film-substrate thermal expansion mismatch. Using a dedicated digital analysis of electron microscopy images, we show that the films contain random nanoregions of substantial strain fluctuations: the local strains vary in the range of similar to(0.5-2)% compared with the average strain magnitude of similar to 1%. Because of strong strain polarization coupling in ferroelectrics, fluctuations of strain produce fluctuations of polarization, which are suggested to cause relaxor-like properties in many mismatched heterostructures. (C) 2017 Elsevier Ltd. All rights reserved.