Don't answer any private messages.

What is the item and its value,  they are trying to buy?

I see you are in the States, and asking this on the UK site,  here,  a buyer would not be able to get any email details to send Private messages it may be different in the States.

If not contact CS in the States again and explain this buyer is contacting you privately to make an off eBay sale,  can they be warned / blocked from contacting you.

You'll need to log out of here clear cache and cookies and log into www.ebay.com

@beesandhoneylondon