We present some of the first infrared spectral maps acquired by SPHEREx. These maps, which to our knowledge are the largest of their type ever compiled in the near-infrared, reveal multiple strong lines due to interstellar ices and polycyclic aromatic hydrocarbons (PAHs) throughout the Cygnus X and North American Nebula regions. The maps emphasize the strongest features arising from the 3 μm H2O, 4.27 μm CO2, and 4.67 μm CO lines and the 3.28 μm PAH feature, all of which are detected over large areas with complex and filamentary spatial distributions. The ice absorption maps of H2O and CO2 in particular broadly trace dense, cold, and well-shielded regions across Cygnus X, consistent with the established picture of efficient ice formation in dense molecular clouds. The interstellar ice features are also detected abundantly in diffuse absorption over wide areas. The relative strengths of the H2O and CO2 features vary among different lines of sight, indicating possible differences in local physical conditions or chemical variations. The 3.28 μm PAH emission correlates with the emission from the 7.7 and 11.2 μm features but shows small differences that may trace the grain-size distribution and variations in the ambient UV field. SPHEREx all-sky spectral imaging—only a small fraction of which is showcased in this work—will support numerous science investigations, including the structure of the Galaxy, the physics of the interstellar medium, and the chemistry of stars.