During magnetic reconnection in collisionless space plasma, the electron fluid decouples from the magnetic field within narrow current layers, and theoretical models for this process can be distinguished in terms of their predicted current layer widths. From theory, the off-diagonal stress in the electron pressure tensor is related to thermal non-circular orbit motion of electrons around the magnetic field lines. This stress becomes significant when the width of the reconnecting current layer approaches the small characteristic length scale of the electron motion. To aid in situ spacecraft and numerical investigations of reconnection, the structure of the electron diffusion region is here investigated using the Terrestrial Reconnection EXperiment (TREX). In agreement with the closely matched kinetic simulations, laboratory observations reveal the presence of electron-scale current layer widths. Although the layers are modulated by a current-driven instability, 3D simulations demonstrate that it is the off-diagonal stress that is responsible for breaking the frozen-in condition of the electron fluid.