The extremely young Class 0 object B1b-S and the first hydrostatic core (FSHC) candidate, B1b-N, provide a unique opportunity to study the chemical changes produced in the elusive transition from the prestellar core to the protostellar phase. We present 40"x70" images of Barnard 1b in the ^13^CO 1-0, C^18^O 1-0, NH_2_D 1(1,1)a-1(0,1)s, and SO 3(2)-2(1) lines obtained with the NOEMA interferometer. The observed chemical segregation allows us to unveil the physical structure of this young protostellar system down to scales of ~500au. The two protostellar objects are embedded in an elongated condensation, with a velocity gradient of ~0.2-0.4m/s/au in the east-west direction, reminiscent of an axial collapse. The NH_2_D data reveal cold and dense pseudo-disks (R~500-1000au) around each protostar. Moreover, we observe evidence of pseudo-disk rotation around B1b-S. We do not see any signature of the bipolar outflows associated with B1b-N and B1b-S, which were previously detected in H_2_CO and CH_3_OH, in any of the imaged species.The non-detection of SO constrains the SO/CH_3_OH abundance ratio in the high-velocity gas.