Ivy supports multiple actions for the selected candidate. You can add your own actions, but by default there is a copy action that may be exactly what you need.
With the default key bindings, you can use
M-o in the minibuffer to bring up the ivy action menu, and
w to copy the currently selected candidate.
I see this does not quite address your use case, since the current file name is not the default candidate for
find-file when ivy is enabled. I always used to use
C-x C-v) as a quick way to grab the current file name, and this works fine with ivy using
M-o w as noted above. (Now I just have a separate key binding to copy
find-fileto copy the current file name, and how does Ivy prevent doing that?
find-fileis useful for copying file name of current buffer. On UP key default completion brings current buffer file name. Then you can select it.