“At this stage of the enquiry we have nothing to indicate this was a terrorist attack,” police said following the attack around 4:30 pm near the bus station of the city of some 100,000 near Stuttgart.
The statement added that the man, who was arrested, “had a dispute” with the woman and killed her “with a machete” before injuring a second woman and a man.
The attacker was “known to police”, it said.
“According to the information available, the perpetrator acted alone, the people of Reutlingen and its surroundings are very probably not in danger,” the statement added.
News channel NTV said there were scenes of panic in the city centre following the attack, which came just two days after a German-Iranian teenager killed nine people and injured 19 others in Munich, Germany's third-largest city, before committing suicide.
The 18-year-old Munich attacker is believed to have been “obsessed” with mass killers such as Norwegian fanatic Anders Behring Breivik and had no links to the Islamic State jihadist group.
The carnage in Munich came just four days after a teenage asylum seeker went on a rampage with an axe and a knife on a regional train near the southern city of Würzburg, injuring five people.
German authorities said the Würzburg attacker was believed to be a “lone wolf” who was “inspired” by Islamic State without being a member of the network.