Looks more like a modeling issue than an Arnold one. You say the surface is flat and it probably is. Do all the faces have the same 'Smoothing Group?' How about the UVs. Do you have a nice square planar UVW on the face?
Sometimes when a mesh has a ton of edges really close together, like yours does between holes in your face, you get artifacts like what you are seeing. It's just too many parallel edges and it doesn't render cleanly.
I've had situations like this and the only way I could solve it was to remodel the face. You might try detaching the face and then using the 'Retopology' modifier to rebuild it. If your face's outer edges and hole edges don't move, they will stay clean. That way the new pure poly face will render the anisotropy correctly.
Rob Holmes

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