Answer
Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
There is no evidence in Shari`ah that forbids women from driving. Women in the lifetime of the Prophet (peace and blessings be upon him) used to ride camels and horses.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
I do not see how we can stop women from driving when we know from the sources that during the Prophet’s time they used to ride camels and horses. Riding horses, then, was not different from driving a Ferrari or sports car today.
For more, see these fatwas:
Are Women Allowed to Work in Islam?
Can Muslim Women Work in Media?
Can Muslim Women Work at Beauty Salon?
Can Men and Women Work Together in Islam?
Allah Almighty knows best.