We have undertaken a systematic search for candidate supernovae from high-redshift Population III stars in a field that has been observed with repeated imaging on a cadence of 2-3 weeks over a 2.2 year baseline, the Spitzer/IRAC Dark Field. The individual epochs reach a typical 5σ depth of 1 μJy in IRAC Channel 1 (3.6 μm). Requiring a minimum of four epochs coverage, the total effective area searched is 214 arcmin2. The unprecedented depth and multi-epochal nature of these data make it ideal for a first foray to detect transient objects which may be candidate luminous pair-instability supernovae from the primordial metallicity first stars. The search was conducted over a broad range of timescales, allowing for different durations of the putative candidates' light-curve plateau phases. All candidates were vetted by inspection of the Spitzer imaging data, as well as deep Hubble Space Telescope/Advanced Camera for Surveys F814W imaging available over the full field. While many resolved-source objects were identified with Spitzer variability, no transient objects were identified that could plausibly be identified as high-redshift supernovae candidates. The resulting 95% confidence level upper limit is 23 deg-2 yr-1, for sources with plateau timescales under 400/(1 + z) days and brightnesses above ~1 μJy.