I am leaning towards no, but would like confirmation and perhaps an example to illustrate.
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The short answer is no. In English, the phones [ɪ] and [i] are not just allophones of a single phoneme.
There are many minimal pairs like "bit-beat", "shit-sheet", "bitch-beach" that establish that [ɪ] and [i] (in a usual English accent) are not always allophones of the same phoneme. To account for the contrasts between these words, we need to define at least two separate phonemes, /ɪ/ and /i/.
However, there are certain specific phonological contexts in which the phonemic contrast between /ɪ/ and /i/ can be analyzed as "neutralized", and for some speakers, there may be allophonic variation of some kind between the phones [ɪ] and [i] in these circumstances.
How to analyze neutralization is a notorious problem for phonological analyses that make use of phonemes, but we could say that in these specific neutralization contexts [ɪ] and [i] are both allophones of some single phoneme, which we could identify as either /ɪ/ and /i/ (or if you use the concept of "archiphonemes", /I/).
A well-known example in English of a vowel that can be analyzed as the neutralization of /ɪ/ and /i/ is the "happy" vowel: in a specific phonological context (word-finally or before another vowel, in a fully unstressed syllable) we can have either [ɪ] or [i], with no contrast. The "happy" vowel is actually not a great example of allophony per se because many contemporary accents only allow it to be realized as the phone [i]. It's more of an example of "diaphonemic" analysis. That said, I have read that there are some British accents, current or historical, where individual speakers show some type of free or conditioned variation between [ɪ] or [i] for the "happy" vowel, which would qualify as actual allophony.
In certain specific North American accents, in particular, there are other contexts where [ɪ] and [i] could be considered to be allophones.
Before /ŋ/, there is no contrast between [ɪ] and [i]. Most accents of English only have [ɪ] in this context, but some North American accents allow (or perhaps even require) [i] instead. Phonetically intermediate realizations of the vowel are also known to exist. So [i] and [ɪ] could be said to be allophones of the same phoneme in this particular position for English speakers.
There have been a number of questions about this topic asked earlier on this site; I linked to some beneath Why is /ɪŋk/ used with "ink" words when the actual pronunciation is /ijŋk/?
Before /r/, many (I think most) North American English speakers have merged /ɪ/ and /i/ to a single vowel phoneme, the realization of which is variable and can range between [ɪ] and [i]. So for speakers with this merger, [ɪ] and [i] could be called "allophones" in this context.