Gagging is a protection mechanism of the body to prevent either objects accidentally entering the throat area (i.e., without the intention to swallow them) or trying to swallow too large objects.
It is entirely possible that the pinky-nail sized piece of banana is still too large for your daughter to swallow without chewing. Another possibility is that her gag reflex is too sensitive. If you suspect the latter case, then I recommend you see a doctor.
Chewing is a learned activity that a child typically learns by imitating their parents. If your child is gagging in their food because they try to swallow it whole without chewing, then you might have to more actively teach them that activity.
Try to both eat something, where you chew your food very explicitly (and maybe in an exaggerated way) and you encourage them to do the same with their food, even if it is the same puree that goes down well without chewing. Once they get the idea that food needs to be chewed, you can do the same game with small pieces of soft solids, like banana.