Police have confirmed that the attacker is a 21-year-old Syrian who is applying for asylum.
The deceased victim is a Polish woman who's been working in the same kebab restaurant as him.
http://www.presseportal.de/blaulicht/pm/110976/3386419
http://www.n-tv.de/panorama/Mann-toe...e18265451.html
EDIT: More information
The attacker is previously known to police
Before the attack the man got into an argument with the woman in a kebab restaurant
After killing the woman he attacked two other people and injured them
The attacker then ran after a police car in a frenzy before being mowed down by a BMW
The deceased victim is a pregnant woman who was working as a cleaner in a kebab restaurant (unclear if it's the same)
The BMW driver who mowed down the attacker is reported to be the son of the kebab restaurant owner
Police are not seeing any hints for a terrorist attack and treating it as a dispute gone sour
EDIT2 (thanks /u/dukenPL): The killed woman is a Polish citizen, a work colleague of the Syrian. Both were employed at the kebab restaurant the dispute started in. According to another co-worker, the Syrian had fallen in love with her, reports Stuttgarter Zeitung