  While ferromagnets have been known and exploited for millennia,
antiferromagnets (AFMs) were only discovered in the 1930s. The elusive nature
indicates AFMs' unique properties: At large scale, due to the absence of global
magnetization, AFMs may appear to behave like any non-magnetic material;
However, such a seemingly mundane macroscopic magnetic property is highly
nontrivial at microscopic level, where opposite spin alignment within the AFM
unit cell forms a rich internal structure. In topological AFMs, such an
internal structure leads to a new possibility, where topology and Berry phase
can acquire distinct spatial textures. Here, we study this exciting possibility
in an AFM Axion insulator, even-layered MnBi$_2$Te$_4$ flakes, where spatial
degrees of freedom correspond to different layers. Remarkably, we report the
observation of a new type of Hall effect, the layer Hall effect, where
electrons from the top and bottom layers spontaneously deflect in opposite
directions. Specifically, under no net electric field, even-layered
MnBi$_2$Te$_4$ shows no anomalous Hall effect (AHE); However, applying an
electric field isolates the response from one layer and leads to the surprising
emergence of a large layer-polarized AHE (~50%$\frac{e^2}{h}$). Such a layer
Hall effect uncovers a highly rare layer-locked Berry curvature, which serves
as a unique character of the space-time $\mathcal{PT}$-symmetric AFM
topological insulator state. Moreover, we found that the layer-locked Berry
curvature can be manipulated by the Axion field, E$\cdot$B, which drives the
system between the opposite AFM states. Our results achieve previously
unavailable pathways to detect and manipulate the rich internal spatial
structure of fully-compensated topological AFMs. The layer-locked Berry
curvature represents a first step towards spatial engineering of Berry phase,
such as through layer-specific moir\'e potential.
