There is no so called short i (/ɪ/) in French. The French i sound is short, of length approximately that of i in “pick”, but of the very same quality of English or american double e as in “weed”, or better “wheat” but shorter still than in this latter (the /i:/ sound is shorter in “wheat” than in “weed”).

No, French (aka international French on Wikipedia or standard French according to others) does not have the /I/ sound as in kit and bit. The French grapheme i (written letter) as in petite, is pronounced like /i:/. There are other realizations of the i grapheme as well.

That’s why French speakers (and Spanish and Portuguese speakers too) cannot make the difference (unless taught or have a really good ear) between the minimal pairs like ship/sheep or bit/beat/beet or chip/cheap.

Whoever said French has that sound was misinformed. The /I/ sound is not “unstable” in English. Say minute, that letter i and that u are both /I/. And the /I/ sound exists in all varieties of English.

“In English, both in Received Pronunciation and in General American, the IPA phonetic symbol /ɪ/ corresponds to the vowel sound in words like “kit” and “English”. It is one of the two vowel sounds we use in English for unstressed syllables, the other one being /ə/.1

In some dictionaries the vowel of KIT is written /i/. There is no confusion as long as the user knows the symbol for /iː/ (the vowel of FLEECE).”

English phonemes

French has the vowel sound /i:/ as in petite. The letter i is pronounced ee in some words as in beet/heat.

And here’s the cutest guy in the world explaining it:

Huito répond pil-poil à la question dans Tutos avec Huito

No, standard French does not have the vowel /ɪ/ (near-close front unrounded vowel), which is the English “short i”. The vowel which is normally written with the letter I in French is a close front unrounded vowel, API symbol /i/. Its realization [i] is fairly stable across French speakers, at least in Europe.¹ Some Canadian speakers do pronounce [ɪ] in closed syllables.

The close vowel exists in English, but only as a long vowel [i:]. However, to French ears, the short, near-close vowel [ɪ] sounds so similar that many French speakers pronounce it as a short, close [i]. If French people pay attention, they’ll perceive the near-close [ɪ] as between the close [i] and the close-mid [e] (and I think that’s how it’s taught in French schools).

Conversely, English speakers might pronounce the letter I as [ɪ] (near-close) instead of [i] (close) when it’s unstressed. French speakers might not even notice. Note that French does not have phonemic (i.e. meaningful) variations on stress or length: stress comes solely from the word and sentence structure, and unstressed speech sounds boring but does not hurt comprehension. So if you’re trying to pronounce French natively and your native language has meaningful stress, try to separate how you pick the vowel quality from whether it’s stressed.

You may find the Wikipedia articles on English and French helpful. It’s difficult for a layman to really understand what all these variations are, but most phonemes have an audio sample, and it at least gives a sense of what is (near-)identical or similar across languages.

¹ _{A small minority pronounce it in a more rounded way, sort of halfway towards [u], but this is nonstandard.}

The French generally spoken in France does not have [ɪ] either phonemically or phonetically, and to my knowledge no variety of French would use it for the first vowel in « s’il vous plait » (though the /l/ often disappears, leaving a shorter first syllable).

To be clear — as you’ve since acknowledged — the terms "short" and "long" that we learn in elementary school in English are misleading when talking about vowels since they’re not really related to length. So we can avoid that red herring of a path.

Canadian French¹

To round out the standard answer above, some varieties of French do have this sound. The one I know best is Canadian French, in which [ɪ] is one of a series of lax realizations of the high vowels.

These are the high vowels in French:

• /y/ : allophones [y] (only option in France) and lax [ʏ]
• /u/ : allophones [u] (only option in France) and lax [ʊ]
• /i/ : allophones [i] (only option in France) and lax [ɪ]

(Note that the exact realization varies by dialect. For [ɪ], I’ve heard [ɨ] and even a diphthong like [ɪ^j].)

These novel lax variants appear in closed syllables, i.e. syllables where the coda is filled by a consonant. Hence, you encounter paradigms like this:

citer [siˈte] ~ cite [sɪt]

lutter [lyˈte] ~ lutte [lʏt]

router [ʁuˈte] ~ route [ʁʊt]

According to Survenant’s research in his answer, the contexts are even more limited: these lax variants only appear in closed final syllables. This isn’t consonant with my own experience, but I bring it up in case I’m mistaken.

Where do these lax variants come from?

It’s sometimes thought that [ɪ] arises in Canadian French because it occurs in English, and Canadian French is often accused of being influenced by English. This deserves a quick comment. While lexical, morphological, and syntactic borrowing across languages is common, phonetic borrowing is rare by comparison. Even heavy exposure to a non-native language rarely penetrates a person’s phonetics; hence, accent often goes unmastered even by proficient learners.

Moreover, the influence of English would not account for [ʏ], which is not present in English; nor the distribution of [ɪ] and [ʊ] only in closed syllables, which is not the case in English; nor the existence of parallel changes in Belgian French, as mentioned by Greg.

Luckily, we have a better explanation for the appearance of these lax vowels. The pattern in which they show up actually exists in standard French. However, in standard French it only applies to the mid vowels, namely the pairs [o] ~ [ɔ], [e] ~ [ɛ], and [ø] ~ [œ]. By adding the set of high vowels, Canadian French extrapolates the pattern to new cases. This type of linguistic change is called "analogy", and it’s relatively common. So this is a plausible hypothesis.²

_{¹ A reasonable component of any answer about "French". If someone asked whether to pronounce /r/ at the end of a syllable in English, the answer would be "No in most British dialects, yes in most American dialects." If someone asked whether Christians baptize infants, the answer would be "Catholics do, most Protestants don’t." Only by demoting one kind or the other could you reduce the answer to "yes" or "no", and from a descriptive linguistics point of view that’s not an interesting exercise.}

_{² Incidentally, Canadian French is also distinguished by certain words with a final consonant absent in other varieties of French, including frette (< frais ?), litte (< lit), icitte (< ici), and toute where you’d expect tout. One could speculate about a symbiosis between this phenomenon and the extra lax vowels. If you create more contexts where a distinctive dialect feature appears, and this feature renders those contexts more salient, is that subconsciously leaning into your sociolinguistic identity?}

Quebec French (French: français québécois; also known as Québécois French or simply Québécois) is the predominant variety of the French language in Canada, in its
formal and informal registers. (Wikipedia, Quebec French
article; see also Canadian French and this answer)

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Tense vowels (/i, y, u/) are realized as their lax ([ɪ, ʏ, ʊ])
equivalents when the vowels are both short (not before /ʁ/, /ʒ/, /z/
and /v/, but the vowel /y/ is pronounced [ʏː] before /ʁ/) and only in
closed syllables. Therefore, the masculine and feminine adjectives
petit ‘small’ and petite ([p(ø)ti] and [p(ø)tit] in France) are
[p(œ̈)t͡si] and [p(œ̈)t͡sɪt] in Quebec. In some areas, notably Beauce,
Saguenay–Lac-Saint-Jean, and (to a lesser extent) Quebec City and the
surrounding area, even long tense vowels may be laxed. (Wikipedia,
Quebec French phonology article: listen to it.)

Therefore this ⟨ɪ⟩ (near-close front unrounded vowel) exists in French for some contexts with some varieties of the French language. Whether one chooses to use it or not if of no consequence whatsoever and is irrelevant.

Le français québécois, aussi appelé français du Québec ou
simplement québécois, est la variété de la langue
française parlée essentiellement par les francophones du
Québec. (Wikipédia, article Français québécois ; voir aussi
Français canadien et cette réponse.)

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Les voyelles /i/, /y/ et /u/ subissent la règle de relâchement ([ɪ, ʏ,
ʊ]) en syllabe fermée lorsqu’elles sont en fin de mot : « mur » se
prononce [mʏːʁ] mais « emmuré » se prononce [ɑ̃myʁe]), « six » se
prononce [sɪs] mais « système » se prononce [sistɛm], « lune » se
prononce [lʏn] mais « lunatique » se prononce [lunatɪk] et « route »
se prononce [ʁʊt] mais « dérouté » se prononce [deʁute]. (Wikipédia,
article Prononciation du français québécois :
l’entendre.)

Donc ce ⟨ɪ⟩ (voyelle pré-fermée antérieure non arrondie) existe en français dans certains contextes avec certaines variétés du français. Qu’on choisisse ou non de l’employer est absolument sans conséquence et hors propos.

I have read all the answers but I would like to add something (I am a French speaker).

When you read the French letter “i”, indeed it´s not the same pronunciation as in english.
BUT, the sound exists in certain graphic environments, for example in the word “laïc”.
“aï” will have the same sound as the english letter “I”.
ï (with umlaut) means you have to sound the first vowel (here “A”), THEN the second letter (here “i”).
So at the end “aï” is pronounced the same as English I.