Although /'pɜ:fɪkt/ and /'ba:skɪt/ is what I have found in my British
dictionaries, I get the impression that the vowel pronounced by most
native speakers of English, no matter where they live, is not the /ɪ/.
The first vowel in /ɪk'spres/ confuses me, too. Are my doubts
I think the point here is that English has two separate vowel systems, the strong system and the weak system. The weak vowels are much less tightly defined than the stressed ones. Although people equate the weak vowel of 'perfect' and 'basket' with /ɪ/, it may vary towards [ə] more than a stressed /ɪ/ would.
The Longman Dictionary of Contemporary English uses a special double-decker symbol to represent a weak vowel that may be [ɪ] or [ə] or something intermediate between them (see picture).
The word perfect is a special case, because in this word some speakers do not weaken the unstressed vowel, but keep it as /e/ (as in the corresponding verb, /pəˈfekt/, where the vowel in question is stressed and therefore strong).
Nevertheless, my advice to Giuseppe is that he can safely treat the unstressed vowels in perfect (noun), basket, and similar words as being just like /ɪ/ (as in KIT). That’s how I believe I pronounce that vowel myself, and that’s what I hear millions of other people say.
A note of warning: Italian learners often have trouble perceiving and making the difference between English /iː/ (FLEECE) and /ɪ/ (KIT). The unstressed vowel we are discussing is certainly not like English /iː/.