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In this paper, we describe a uniform analysis of eight transits and eleven secondary eclipses of the extrasolar planet GJ 436b obtained in the 3.6, 4.5, and 8.0 μm bands using the IRAC instrument on the Spitzer Space Telescope between UT 2007 June 29 and UT 2009 February 4. We find that the best-fit transit depths for visits in the same bandpass can vary by as much as 8% of the total (4.7σ significance) from one epoch to the next. Although we cannot entirely rule out residual detector effects or a time-varying, high-altitude cloud layer in the planet's atmosphere as the cause of these variations, we consider the occultation of active regions on the star in a subset of the transit observations to be the most likely explanation. We find that for the deepest 3.6 μm transit the in-transit data have a higher standard deviation than the out-of-transit data, as would be expected if the planet occulted a star spot. We also compare all published transit observations for this object and find that transits observed in the infrared typically have smaller timing offsets than those observed in visible light. In this case, the three deepest Spitzer transits are all measured within a period of five days, consistent with a single epoch of increased stellar activity. We reconcile the presence of magnetically active regions with the lack of significant visible or infrared flux variations from the star by proposing that the star's spin axis is tilted with respect to our line of sight and that the planet's orbit is therefore likely to be misaligned. In contrast to the results reported by Beaulieu et al., we find no convincing evidence for methane absorption in the planet's transmission spectrum. If we exclude the transits that we believe to be most affected by stellar activity, we find that we prefer models with enhanced CO and reduced methane, consistent with GJ 436b's dayside composition from Stevenson et al. It is also possible that all transits are significantly affected by this activity, in which case it may not be feasible to characterize the planet's transmission spectrum using broadband photometry obtained over multiple epochs. These observations serve to illustrate the challenges associated with transmission spectroscopy of planets orbiting late-type stars; we expect that other systems, such as GJ 1214, may display comparably variable transit depths. We compare the limb-darkening coefficients predicted by PHOENIX and ATLAS stellar atmosphere models and discuss the effect that these coefficients have on the measured planet–star radius ratios given GJ 436b's near-grazing transit geometry. Our measured 8 μm secondary eclipse depths are consistent with a constant value, and we place a 1σ upper limit of 17% on changes in the planet's dayside flux in this band. These results are consistent with predictions from general circulation models for this planet, which find that the planet's dayside flux varies by a few percent or less in the 8 μm band. Averaging over the eleven visits gives us an improved estimate of 0.0452% ± 0.0027% for the secondary eclipse depth; we also examine residuals from the eclipse ingress and egress and place an upper limit on deviations caused by a non-uniform surface brightness for GJ 436b. We combine timing information from our observations with previously published data to produce a refined orbital ephemeris and determine that the best-fit transit and eclipse times are consistent with a constant orbital period. We find that the secondary eclipse occurs at a phase of 0.58672 ± 0.00017, corresponding to ecos (ω) = 0.13754 ± 0.00027, where e is the planet's orbital eccentricity and ω is the longitude of pericenter. We also present improved estimates for other system parameters, including the orbital inclination, a/R⋆, and the planet–star radius ratio.