No. Those were done away with years ago.
Best you can do is set up a listing with some safeguards against someone else buying it. I recommend a fixed-price listing with a high Buy It Now price with Immediate Payment Required and a Best Offer option. In order for someone else to block the sale to the intended buyer s/he would have to actually pay the high price. The intended buyer can make a Best Offer (send him/her the item number as soon as you get it, and instructions) which you can accept before it expires (the buyer might have to pre-authorize automatic payment to make the offer depending on whether you have that option enabled in your Buyer Requirements page). Reject or ignore offers from anyone else.
ETA: The term "Private Listing" means something entirely different: a listing where bidders'/buyers' identities are even more protected than the a***b aliases for normal listings.
