We present 850 mu m imaging polarimetry data of the rho Oph-A core taken with the Submillimeter Common-User Bolometer Array-2 (SCUBA-2) and its polarimeter (POL-2) as part of our ongoing survey project, B-fields In STar forming RegiOns (BISTRO). The polarization vectors are used to identify the orientation of the magnetic field projected on the plane of the sky at a resolution of 0.01 pc. We identify 10 subregions with distinct polarization fractions and angles in the 0.2 pc rho Oph-A core; some of them can be part of a coherent magnetic field structure in the rho Oph region. The results are consistent with previous observations of the brightest regions of rho Oph-A, where the degrees of polarization are at a level of a few percent, but our data reveal for the first time the magnetic field structures in the fainter regions surrounding the core where the degree of polarization is much higher (>5%). A comparison with previous near-infrared polarimetric data shows that there are several magnetic field components that are consistent at near-infrared and submillimeter wavelengths. Using the Davis-Chandrasekhar-Fermi method, we also derive magnetic field strengths in several subcore regions, which range from approximately 0.2 to 5 mG. We also find a correlation between the magnetic field orientations projected on the sky and the core centroid velocity components.