Just spent some time going through the Victoria Golf Club 100th anniversary publication. As Gary Slatter said below, if "west of the Mississippi" is considered to be within the continental US, Canadian clubs would be excluded. In that case, so far as I am aware, the honour would belong to Gearhart Golf Club in Oregon. As the phrase, "west of the Mississippi" does not seem to consider national boundaries, I think the answer is truly Victoria Golf Club in Victoria BC.
I have pulled out a copy of their centennial publication "Victoria Golf Club 1893-1993" which I picked up in a used bookstore near Victoria a few years ago.
Chapter 1, headed "Beginnings" starts off:
""most people find it difficult to imagine the City of Victoria, or indeed any other intelligent community, existing without a really servicable golf course; yet so it was until the Victoria Golf Club came into existence on November 7, 1893. Furthermore, we have reliable evidence that it is the second-oldest golf club in its origianl location in North America (Shinnecock Hills, founded in New York State in 1892 is the oldest). To go even further back, one authority claims that a small group of Hudson's Bay officers were the first to play golf in the Pacific Northwest in the 1840s. An unnamed factor at Steilacoom, one of the HBC farms administered by Nisqually, on Puget Sound, inwhat is now Washington State, sent for his clubs from Scotland, and a small, seven hole course was laid out some two miles from the present village of Steilacoom.
Unfortunately, this brave venture came to an end when Washington became a US Territory and Steilacoom a military fort. . . . . "
The book goes on to point out a few other areas on Vancouver Island where golf was played at what is now Beacon Hill Park in Victoria in 1889 but no club was formed and the area eventulally became a public park, no golf.
Another reference I found in this area was referred to in the Victoria Golf Club history is to a publication "The Earliest History of Golf in Vancouver, BC" by Francis Chaledcott. It goes on to speak of the Vancouver Golf Club which was formed in 182 at Jericho, which is now a park / residential area on the harbour in Vancouver. Vancouver Golf Club is now located (and has been for a heck of a long time) in Coquitlam, which is a suburb to the east of Vancouver.
Enough. The Victoria Golf Club book is a great read. Chapter 2, entitled "Serious Progress" opens with a quote, " . . the first, the really basic rule of golf is never to let business, mere commercial or professional employment interest, interfere with the game . . " which is a quote taken from the Tacoma Legend.
Bob Jenkins