A Willybat is a small, australian creature crossed between a bat and the male genitalia. It lives in caves and feeds on small Iranian worms.
They are very friendly and make great pets. They are extremely playful and like frequent petting.
From this creature the term willybatting around has also become a wildly used phrase. It would suggest someone has lack of phocus and is behaving in an erratic, over playful way.
"Hey look, there flys a Willybatt"
"Ow i had to take me Willybat to the vet today, it keeps falling out of the sky, I think it's wing is broken"
"Stop willybatting around"