So in summary, and
trying to get some help here gentlemenWe don't know if this was a "rifle" or a L.C. Smith Maker, Syracuse shotgun. The hammerless was introduced in 1886.
The Critic (Washington, District of Columbia), Friday November 12,
1886Miss Annie Oakley, the sharp shooter, possesses the most valuable rifle in America. Every screw in it is of solid gold, and a figure of herself, also of gold, is set in the stock. It was a present from L.C. Smith of Syracuse.
Annie's letter on p.1 stated 4 Lancasters made in
1887The
1894 New York Times piece above listed 2 Lancaster, 1 Scott, 1 Parker & 1 Francotte.
The Art of Wingshooting, William Bruce Leffingwell,
1895https://books.google.com/books?id=e34EmE3tkfkC&pg=PA106&vq “I was permitted to examine many of the fine guns used by Miss Oakley in exhibition shooting, and noticed among them one Charles Lancaster ejector, one Charles Lancaster non-ejector, a Cashmore hammerless,
a magnificent Smith ejector with a gold figure of herself inlaid, a Parker
hammerless, a Scott Monte Carlo, a Scott ejector of highest quality, and an exquisite little Francotte ejector with Whitworth barrels.
Could that be the 1886 gift from L.C. Smith??This claims that FIVE Cashmores were made for Annie
http://www.jbssoftware.com/cgi-bin/mnlist.cgi?condon/24875 The 2 Parker hammer guns on p. 2 have clear provenance. Here is one
https://centerofthewest.org/2014/05/25/points-west-annie-oakley-collecction/ Where is the 1895 referenced Parker hammerless??Her Tiffany engraved possibly Smith A1 was made 1898 or 1899. Dr Jim and I have contacted both the BBHC and the Gene Autry museums hoping to get the SN so the Hunter Arms' records can be checked.
Calling this Gold Hibbard an "Annie Gun" is a bit of a stretch
http://www.nramuseum.org/guns/the-galleries/the-american-west-1850-to-1900/case-21-tin-stars-sixguns-frontier-justice/annie-oakley's-hibbard-shotgun.aspx
(You'll need to cut and paste the link)
Here is another
https://garyhendershott.net/western-catalog.php 
