RMB "Surface Bodies" Folder and click "Insert in to a new part..."
Go to the surface bodies folder in feature manager and expand it. Right-click on the surface body that you want to export and select: Insert into a new part. You'll be prompted to specify a solidworks part file name
then in the part where you want to insert the feature, go to insert==>part and browse to the part you just created.
That will get the surface into your other part. If the original surface updates, the change will propogate to this surface.
If you want to skip creating an intermediary file, you can insert the source part into your target part and have it only include the surface bodies. This also maintains a link to the source document.
In either case, you can use move-copy bodies to position the surface on your new part whereever you want.
If you want to use the positioning of an assembly, you can edit your source component and use knit surfaces or offset surface (with a distance of zero), pick the faces that comprise you're surface body and those will come into your target part in their same relative location. If the source part moves or changes, the destination part will update. If you don't want the destination part to automatically update, you can break the external references.
You can also export a surface to an exchange format like step or iges and insert that into another part. This doesn't create external references.
Let me know if you need more information about any of these techniques.
I'm having a similar problem that I'm not solving at all even if I try to follow your steps.
I supose I'm missing something basic, but I don't know what else to try... I explain:
While working into an assembly file (SW2016), I am editing a part that needs to be cutted by a surface offset of another part. So what I need is to copy the offset surface from the original part to the cutted (not yet) one. Now I'm describing the steps I do:
1. In the feature tree right click in the part where the offset surface is > Edit part.
2. In the surface body folder of that part I find my beloved surface and I right click on it > Insert into a new part.
When it's time to save on the prompt window I can only save the surface being a new part itself, if I choose the part where I want to add the surface directly through the prompt browser I have an error after the piece is opened as a new file in the SW session ("Error while saving the document"). If I don't choose a file with the browser and click OK I also have the surface opened as a no-named new part and I am only able to save it as a new part (obviously if I choose the surface-needing part I am asked to overwrite it, but never to add or combine both parts). So... I save my surface as a new part to try to add it to the other part.
3. As I read in your instructions, I go out of the "editing part mode" and click insert > part and into the prompt I choose the new part that contains only the surface. I'm not finding a merge, combine, add to part or similar option, so I can only add this part as a new one into the assembly as the tree shows after the part incorporation.
4. I still having the same problem: My cutting surface is into a different part to the one that needs to be cutted.
So my question is:
Between steps 2 and 3, how can I save the surface into the target part file instead of rewrite it?
What am I missing on your posted process?
Thank you so so much!!