Can you say "elle m'a tendu le doigt" instead of "sa doigt" because you know it's her finger? I thought when it's obvious to whom the body part belongs, the French prefer not to specify as in "J'ai mal à la tête."
Freeform Writing Exercise C1
I suspect the use of tendre son doigt has achieved idiomatic status. Note, however, that you still use possessive pronouns when there is special emphasis on the body part. See also: Special cases when you use mon/ma/mes/etc with parts of the body (French Possessive Adjectives)
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