In a letter from Baranov to Shelikhov, written in July 1793, he mentions an English ship named Phoenix. It doesn't explicitly mention when the ship arrived there or if it had wintered there.
Here I encountered the English ship Phoenix, 2 masts, 85 feet long. This ship has sailed from East Indian to Canton, Manila, and then to Nootka in America, and from there followed the coast to Chugach Bay. On this voyage, trading with different people, they accumulated a big quantity of furs. During a storm, they lost two masts and now were replacing them in Chugach Bay. Their intention was to sail back to Canton. The captain, by the name of Moore, is an Englishman from Ireland. At first he was afraid of my fleet of baidarkas and the natives, but finally we became friends and during bad
weather I spent five days on board his ship and had all my meals with him.
However, the reply from Shelikhov does indicate that foreign ships were unwelcome
We are very much astonished at your unconcern about the visit of the English ship (Phoenix, Capt Hugh Moore).
You knew, even before you got the new regulations, that visits by foreign vessels cannot be tolerated. … The ship you
met was in serious danger from natives, but you warned them. I don’t think that would have had such friendly assistance
if you happened to be at Nootka or California. Probably we would have had to say farewell to you, and would
never have seen you again….. In the future….be bold enough to tell the foreigners that they have no right to trade.
So this English ship may have been the only one to visit at that time.