The spatial extension of a γ-ray source is an essential ingredient to determine its spectral properties, as well as its potential multiwavelength counterpart. The capability to spatially resolve γ-ray sources is greatly improved by the newly delivered Fermi-Large Area Telescope (LAT) Pass 8 event-level analysis, which provides a greater acceptance and an improved point-spread function, two crucial factors for the detection of extended sources. Here, we present a complete search for extended sources located within 7°from the Galactic plane, using 6 yr of Fermi-LAT data above 10 GeV. We find 46 extended sources and provide their morphological and spectral characteristics. This constitutes the first catalog of hard Fermi-LAT extended sources, named the Fermi Galactic Extended Source Catalog, which allows a thorough study of the properties of the Galactic plane in the sub-TeV domain.