The nearby Large Magellanic Cloud (LMC) provides a rare opportunity of a spatially resolved view of an external star-forming galaxy in gamma-rays. The LMC was detected at 0.1-100GeV as an extended source with CGRO/EGRET and using early observations with the Fermi-LAT. The emission was found to correlate with massive star-forming regions and to be particularly bright towards 30 Doradus. Studies of the origin and transport of cosmic rays (CRs) in the Milky Way are frequently hampered by line-of-sight confusion and poor distance determination. The LMC offers a complementary way to address these questions by revealing whether and how the gamma-ray emission is connected to specific objects, populations of objects, and structures in the galaxy. We revisited the gamma-ray emission from the LMC using about 73 months of Fermi-LAT P7REP data in the 0.2-100GeV range. We developed a complete spatial and spectral model of the LMC emission, for which we tested several approaches: a simple geometrical description, template-fitting, and a physically driven model for CR-induced interstellar emission. In addition to identifying PSR J0540-6919 through its pulsations, we find two hard sources positionally coincident with plerion N 157B and supernova remnant N 132D, which were also detected at TeV energies with H.E.S.S. We detect an additional soft source that is currently unidentified. Extended emission dominates the total flux from the LMC. It consists of an extended component of about the size of the galaxy and additional emission from 3 to 4 regions with degree-scale sizes. If it is interpreted as CRs interacting with interstellar gas, the large-scale emission implies a large-scale population of ~1-100GeV CRs with a density of ~30% of the local Galactic value. On top of that, the 3 to 4 small-scale emission regions would correspond to enhancements of the CR density by factors 2 to 6 or higher (Abridged).Read less <