We present a deep $K^{prime}$-band (2.12$mu$m) imaging of 1arcmin $times$ 1arcmin Subaru Super Deep Field (SSDF) taken with the Subaru adaptive optics (AO) system. Total integration time of 26.8 hours results in the limiting magnitude of $K^{prime} sim 24.7$ (5$sigma$, 0farcs2 aperture) for point sources and $K^{prime} sim 23.5$ (5$sigma$, 0farcs6 aperture) for galaxies, which is the deepest limit ever achieved in the $K^{prime}$ band. The average stellar FWHM of the co-added image is 0farcs18. Based on the photometric measurements of detected galaxies, we obtained the differential galaxy number counts, for the first time, down to $K^{prime} sim 25$, which is more than 0.5 mag deeper than the previous data. We found that the number count slope $dlog N/dm$ is about 0.15 at $22 < K^{prime} < 25$, which is flatter than the previous data. Therefore, detected galaxies in the SSDF have only negligible contribution to the near-infrared extragalactic background light (EBL), and the discrepancy claimed so far between the diffuse EBL measurements and the estimated EBL from galaxy count integration has become more serious . The size distribution of detected galaxies was obtained down to the area size of less than 0.1 arcsec$^2$, which is less than a half of the previous data in the $K^{prime}$ band. We compared the observed size-magnitude relation with a simple pure luminosity evolution model allowing for intrinsic size evolution, and found that a model with no size evolution gives the best fit to the data. It implies that the surface brightness of galaxies at high redshift is not much different from that expected from the size-luminosity relation of present-day galaxies.
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