Hi, why is that after infinitives the verb to have is used?
Tell her.
-I have!
Does it mean: I (already) have. ?
Is it possible to respond: I did!, as in I (already) did. ?
Thanks
Strictly speaking "tell" is not an infinitive here, it's an imperative.
Now, regarding "have", this is a very common English use of ellipsis (your posting suggests you are actually already aware of this). The use of "have" is enough by itself to imply greater unstated meaning.
Think of your example this way and note in [] what is implied:
Tell her.
I have [told her]!
And yes, you can certainly also say:
I did [tell her]!
It's short for "I have told her" (or "I did tell her").