Normals in my model are not animated.

I have a blender model with animation. As you can see, normals are kinda wrong. It looks like I have multiple normals per vextes, so that should be okey. This is how it looks in unity before and after animation.

As you can see, my model should be lit up just like the cube next to it. When I hit apply Armature modifier in blender, my animation is lost but my model looks like this.