Is there a short I vowel sound in the word gift?

No, there is not a short /ɪ/ vowel sound in the word "gift."

The vowel sound in "gift" is a short /ɪ/ sound.

The short /ɪ/ sound is different from the long /i/ sound, which you might be thinking of. The long /i/ sound is found in words like "bike" or "time."