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I'm Italian and I've never in my life seen "î", I wouldn't even know how to read it
According to German Wikipedia it is old spelling and thus, no wonder you didn't come across it:
In Italian, the circumflex used to be used primarily in the pluralisation of words with a final -io to mark the coincidence of two -ii: il principio "the principle" → i principî, in contrast to i principi, the plural of il principe "the prince". In addition to principî, there was also the full spelling principii, which was not pronounced correctly. Today, the words for "principles" and "princes" are spelt principi without distinction.
(translated using DeepL)
According to the English article, it is also used in Emilian and Friulian. In both, a long vovel is indicated with a circumflex.
But I have come across other old spellings, like "j" used in diphthongs in place of "i", like in "jeri" (old spelling for "ieri", "yesterday") or in "naja" (old word for compulsory military service time). So it must be even older/rarer than that, and I would still say "j" it's not an Italian letter because nobody uses it exept to write "Jesolo" but that's a name, not a regular word.
Fun fact: because of the old usage of "j" some text to speech are "broken". The one on railway speakers always reads "RailJet" as "Railiet" which sounds funny.
Also used in some dialects, like Piemontese, where Y and J can be used instead of I, especially in family names