The statement adds: “Not a day went by without Olive singing and dancing her way round the house and spreading her affection; always kissing and cuddling and going out of her way to be kind to everyone.
“Olive’s memory will live on with all the colourful paintings we have around the house. Quite often she would take herself off and come back with a beautiful work of art as she was so creative.
“Olive had two brothers, Teddy and Sonny, who she was inseparable from and looked up to with so much love.”
Olive was due to start primary school and was “so excited” to join one of her brothers there, the statement said.
It adds: “No words can express the loss and pain felt across the family and all that knew Olive, we are utterly devastated to think we can no longer hold her in our arms.”
A vigil is being held at St Lawrence’s Church in Gotham on Friday for friends, family and members of the local community.
Nottinghamshire Police said a 59-year-old man has been arrested and bailed on suspicion of causing death by dangerous driving.
Sgt Paul Clark said: “This is a deeply tragic incident, and our thoughts continue to be with Olive’s loved ones at this unimaginably difficult time.
“The circumstances of her death are being investigated, and we would ask people not to speculate on social media.”