That is not an option. I'm not concerned about formatting the text for a printout for my own purposes or anything. This has to go to a publisher, and they want a no frills document, nothing with styles (which I invariably find too complicated to go into anyway). Just plain text, at most a TAB here and there, italics and bold, is okay. And smallcaps.
But for some reason smallcaps seems to be a stepchild to italics and bold. Whereas the latter are always "remembered" when pasting (unles you paste without any formatting), smallcaps isn't.

It's so weird: I find now that I can retain smallcaps if I choose "use target styles" when pasting. I'm assuming that means the text assumes the outward look of the target text into which I'm pasting.
So this works, smallcaps are retained.
But... the target text has 12 point letters, and the pasted text has 14 point letters. If the option is called "use target styles", then why doesn't the text receive a 12 point attribute?
It's no big problem, it just seems so illogical or counterintuitive.
I have few problems with Word, but this pasting behaviour is really weird.