We investigate the local correlation between the 1.4-GHz radio continuum and 60-mu m far-infrared (FIR) emission within the Large Magellanic Cloud (LMC) on spatial scales between 0.05 and 1.5 kpc. On scales below similar to 1 kpc, the radio-FIR correlation is clearly better than the correlation of the cold gas tracers (CO and H I) with either the radio or the FIR emission. For the LMC as a whole, there is a tight correlation between the radio and FIR emission on spatial scales above similar to 50 pc. By decomposing the radio emission into thermal and non-thermal components, however, we show that the scale on which the radio-FIR correlation breaks down depends on the thermal fraction of the radio emission; regions that show a strong correlation to very small scales are the same regions where the thermal fraction of the radio emission is high. Contrary to previous studies of the local radio-FIR correlation in the LMC, we show that the slope of the relation between the radio and FIR emission is non-linear. In bright star-forming regions, the radio emission increases faster than linearly with respect to the FIR emission (power-law slope of similar to 1.2), whereas a flatter slope of similar to 0.6-0.9 applies more generally across the LMC. Our results are consistent with a scenario in which the ultraviolet photons and cosmic rays in the LMC have a common origin in massive star formation, but the cosmic rays are able to diffuse away from their production sites. Our results do not provide direct evidence for coupling between the magnetic field and the local gas density, but we note that synchrotron emission may not be a good tracer of the magnetic field if cosmic rays can readily escape the LMC.