In the assembly, Edit the part that needs to have the surface (Right Mouse Button on the part in the Assembly Feature Manager, Edit Part)
(The part being edited becomes blue, the other parts become transparent).
Select the surface to be copied.
Insert--->Surface Offset (0mm)
The surface will be copied in the part.
or continue working in the Part being edited.
The arrow next to the "Surface-Offset2->" indicates that the feature is using an external reference.