Au contraire! Where do you think mod_rewrite would find the referralID=12345 (and why in the world would you replace the query string's &'s with $'s?).
You might benefit from reading the mod_rewrite tutorial linked in my signature as it contains explanations and sample code. It's helped may members and should help you, too.
Is it possible to use mod_rewrite to make the URL look like this?
Are you serious? I thought the question was to go in the other direction (possible EXCEPT for the $referralID=12345). When working with SEO redirections, it's the webmaster's task to define the format for the links to be seen then create the mod_rewrite code to convert that to what Apache needs to serve the required page. Your question is "how can I change the visible URI to something which cannot be served?" If you read my tutorial, you'll know that it can be done but not in the manner you suggest and it's a burden on the server to perform loopy redirects without a valid reason (like 301 redirecting old format links to new).