To study the size and strain rate dependency of copper polycrystalline microstructures, a multi-layered copper/Al$_2$O$_3$ thin film was deposited on a Si substrate using a hybrid deposition system (combining physical vapour and atomic layer deposition). High temperature treatment was applied on the "As Deposited" material with ultrafine-grained structure to increase the average grain size, resulting in a "Heat Treated" state with microcrystalline structure. Focused ion beam milling was employed to create square shaped micropillars with two different sizes, that were subjected to compressive loading at various (0.001/s - 1000/s) strain rates. The detected two distinct anomalies in the strain rate sensitivity behavior appearing at high strain rates could be related to the pillar diameter and the grain size of the deformed samples. The Al$_2$O$_3$ interlayer studied by transmission electron microscopy showed excellent thermal stability and grain boundary pinning by precipitation, also resulting in the homogeneous deformation of the pillars and preventing shear localization. Geometrically necessary dislocation densities estimated by high (angular) resolution electron backscatter diffraction presented inhomogeneous dislocation distribution within the deformed pillar volumes, that is attributed to the proximity of the sample edges. Finally, the Al$_2$O$_3$ interlayers successfully suppressed any possible recrystallization processes, contributing to the excellent film stability, that makes the proposed coating ideal to be operating under extreme conditions.