"In the course of the ongoing criminal investigations, there are indications that a possible custody dispute may have been the trigger for the incident," police said.
The three lived in the Passau area, the police added.
The man, a 48-year-old Iraqi citizen, has been detained and is being questioned.
It is unclear whether he has made a statement about the incident.
A decision on whether the suspect will remain in custody is expected on Sunday.
The Passau Criminal Police Inspectorate, together with the Passau Public Prosecutor's Office, has taken over the investigation.
Based on current findings, the allegation is attempted homicide.
The wife and daughter, along with three other injured individuals, are being treated in hospital.
The severity of the injuries is not yet entirely clear but no one is in a life-threatening condition, a police spokesman said.
The relationship between the other injured individuals and the family - if there is one - is not yet known, the spokesman added.
The man drove his car into a group of people on the pavement in a street near Passau's main train station about 3.30pm.
Police quickly assumed that the man had deliberately steered the vehicle into the group.
People in the area alerted the police, and the injured were taken to hospital by ambulance.
The scene of the incident was cordoned off and the car involved removed with a crane in the early evening.
