I have two files opened in vim, each in one tmux pane. I would like to copy let's say 10 lines from one file to another.
How can I accomplish this not using the mouse's
You'll have to use tmux shortcuts. Assuming your tmux command shortcut is the default: Ctrl+b, then:
tmux is quite good at mapping commands to custom keyboard shortcuts.
See Ctrl+b,? for the full list of set keyboard shortcuts.
You could use the system clip board with "*y and "*p instead of the normal y and p.