Um...no...only the R/O sections. Programs have initialized R/W spaces that once written are gone. There'd be no reason to have those sections marked as COW, unless you already had someone sharing the page (like a forked copy). But any unrelated process likely wouldn't use a COW copy, as they'd need the pristine data as it was supposed to be when the program loaded.
Sure, any overview of virtual memory processes this length is bound to be oversimplified in some place. But we can safely ignore this detail as it doesn't differ between dynamically vs. statically linked programs.