We report Spitzer Space Telescope observations during predicted transits of the exoplanet Proxima Centauri b. As the nearest terrestrial habitable-zone planet we will ever discover, any potential transit of Proxima b would place strong constraints on its radius, bulk density, and atmosphere. Subsequent transmission spectroscopy and secondary-eclipse measurements could then probe the atmospheric chemistry, physical processes, and orbit, including a search for biosignatures. However, our photometric results rule out planetary transits at the 200 ppm level at 4.5 μm, yielding a 3σ upper radius limit of 0.4 R⊕ (Earth radii). Previous claims of possible transits from optical ground- and space-based photometry were likely correlated noise in the data from Proxima Centauri's frequent flaring. Our study indicates dramatically reduced stellar activity at near-to-mid infrared wavelengths, compared to the optical. Proxima b is an ideal target for space-based infrared telescopes, if their instruments can be configured to handle Proxima's brightness.