The 48-year-old Iraqi man has been transferred to a correctional facility, they said.
According to initial findings, police suspect a custody dispute may have been the motive.
The man and his wife are reportedly separated, the police said.
"There had already been reasons for police action in connection with his wife in the past," a police spokesman said.
Three of the people injured in the incident, including the daughter, have already been released from hospital.
The 40-year-old wife and a 39-year-old woman are still receiving treatment but their lives are not in danger, police said.
The two other people injured in the incident were an 18-year-old woman and a 52-year-old woman, who suffered abrasions and bruises.
Their relationship to the family is not known.
The man drove his car into the group of people near the southern German city's main train station about 3.30pm on Saturday.
Passers-by alerted the police, and the man was detained and questioned shortly afterwards.
It is not known whether he has made any statement about the incident.
