No. Or, to be quite precise, since you can't prove a negative, there's no known work of Basho's that contains such a poem. In fact, while the Narrow Road to the Deep North passes through Matsushima, there are no known Basho haiku at all about the place, although his disciple and fellow traveler Sora did write one.
Instead, the first known appearance of the haiku in question is the collection Matsushima Zushi (松島図誌), edited by Komon Sakurada (櫻田虎門) and first published in the 3th year of Bunsei (1820), over a hundred years after Basho's death in 1694. On spread 35, to the right of the fold, this contains the following poem:
松嶋や さてまつしまや 松嶋や
Matsushima-ya / Sate Matsushima ya / Matsushima ya
This is credited to Tawara-bō (田原坊, "Boy from Tawara") of Sagami Province (相模国, today's Kanagawa, west of Tokyo), an obscure kyoka ("crazy poem") poet known today more or less solely for apparently composing this poem. Change the sate (さて, "now") to aa (ああ, "ah"), use kanji for all three Matsushimas, and you get the canonical version:
松島や ああ松島や 松島や
Matsushima ya / Ah, Matsushima ya / Matsushima ya
And veering into speculation, since Matsushima is associated with Basho but they didn't have any poems, this was popularly adopted as one since it was simple, memorable and of suitably obscure origins.