To add to that, DD's first pets were two pairs of doe rats for her fourth birthday.
Before that, she helped me hand rear the sugar gliders, and had an active role in training three of our four dogs, as well as a few fosters along the way.
Because of our heavy involvement in a very wide variety of animal care, she's grown up with an extremely disciplined and responsible view of pets.
She begged for two years before she got her rats, but I refused until she was old enough to care for little lives herself.
She's young, but our rules are still the same. Forget to feed, water, clean or play, and you lose your rights to have your own pets. It's been almost 2 years and she has not missed a single day caring for them.
I want her to understand the gravity of having total responsibility for another living being and so far she's been just fantastic; I'm very proud. It's a difficult concept for a young child.
Wait until your daughter is older and has more of a personality. If she's naturally mature and responsible with animals, go for it!