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We analyze the redshift- and luminosity-dependent sizes of dropout galaxy candidates in the redshift range z ~ 7-12 using deep images from the 2012 Hubble Ultra Deep Field (UDF12) campaign, which offers two advantages over that used in earlier work. First, we utilize the increased signal-to-noise ratio offered by the UDF12 imaging to provide improved measurements for known galaxies at z ≃ 6.5-8 in the HUDF. Second, because the UDF12 data have allowed the construction of the first robust galaxy sample in the HUDF at z > 8, we have been able to extend the measurement of average galaxy size out to higher redshifts. Restricting our measurements to sources detected at >15σ, we confirm earlier indications that the average half-light radii of z ~ 7-12 galaxies are extremely small, 0.3-0.4 kpc, comparable to the sizes of giant molecular associations in local star-forming galaxies. We also confirm that there is a clear trend of decreasing half-light radius with increasing redshift, and provide the first evidence that this trend continues beyond z ≃ 8. Modeling the evolution of the average half-light radius as a power law, ∝(1 + z)^s , we obtain a best-fit index of s = -1.30^(+0.12)_(-0.14) over z ~ 4-12. A clear size-luminosity relation is evident in our dropout samples. This relation can be interpreted in terms of a constant surface density of star formation over a range in luminosity of 0.05-1.0L^*_(z=3). The average star formation surface density in dropout galaxies is 2-3 orders of magnitude lower than that found in extreme starburst galaxies, but is comparable to that seen today in the centers of normal disk galaxies.