  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$~{\mu}m$, yielding a 3$\sigma$ upper radius limit
of 0.4~$R_\rm{\oplus}$ (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. Follow-up observations should
focus on planetary radio emission, phase curves, and direct imaging. 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.
