“There are dozens of injured, including some who are gravely hurt,” a spokeswoman for Volkmarsen police told AFP.
Speaking to DPA news agency, a police spokesman added that children were among the injured. The driver has been arrested.
Germany is on high alert following a shooting spree by a far-right gunman in the city of Hanau, also in Hesse, last Wednesday, that left 10 people dead.
Monday's incident took place as residents in many parts of the country celebrate Rose Monday, a highlight of annual carnival festivities that sees adults and children alike dress up and attend parades where people play music and throw candies from floats.
Police in Hesse announced on Twitter that all carnival parades across the state had been cancelled as a precaution.
A spokeswoman for police in the town of Volkmarsen where the incident took place told AFP “it is too soon” to say whether the driver ploughed into the crowd on purpose.
Volkmarsen is situated in the central state of Hesse, northwest of the city of Kassel.
Pictures from the scene showed police officers and rescue vehicles next to a silver Mercedes hatchback with its doors open, having apparently come to a halt outside a REWE supermarket.
According to witness reports in local media, the incident in the town of some 7,000 people started at around 2:30 pm.
“Eyewitnesses say the driver bypassed a street closure and raced into the crowd at full speed,” the Frankfurter Rundschau newspaper wrote.
A pile of debris is seen on the side of the road next to the car, including a knocked-over traffic cone and bottles of sparkling wine.
Several dozen people are seen milling around on the sidewalk, many in colourful costumes.
“We're on the site with a big squad. An investigation is underway,” tweeted local police after the incident.
Der Fahrer wurde festgenommen.
Wir sind mit einem großen Aufgebot vor Ort. Die Ermittlungen zu den Hintergründen laufen.
Sobald gesicherte Informationen vorliegen, veröffentlichen wir diese hier.
— Polizei Nordhessen (@Polizei_NH) February 24, 2020
The incident comes as Germany is still reeling from a shooting spree in the city of Hanau, in the same German state of Hesse, that left 10 people dead last Wednesday.
The gunman, who left behind a racist manifesto, first opened fire at a shisha bar and a cafe in Hanau, killing nine people, before shooting dead his mother and himself.
The rampage fuelled concerns over Germany's increasingly emboldened far right scene, after a pro-migrant politician was murdered in June and an anti-Semitic attack on a synagogue left two dead in the city of Halle last October.
Condemning the violence in Hanau, Chancellor Angela Merkel said Germany had to fight back against the “poison” of racism and hatred running through German society.
Thousands of Germans later joined vigils to mourn the victims and call for more protection for minorities.
Many also used the occasion to vent their anger at the far-right AfD party, which has been accused of stoking anti-foreigner sentiment and normalising hate speech in recent years.
Germany's deadliest terror attack in recent history took place in 2016 when a jihadist drove his truck into a crowded Berlin Christmas market, killing 12 people.
The attacker, a failed Tunisian asylum seeker, had pledged allegiance to the Islamic State (IS) group.
The Christmas market rampage prompted police across Germany to tighten security at public gatherings.
In response to the shootings in Hanau, Interior Minister Horst Seehofer on Friday again vowed to ramp up security and put more police at mosques, train stations, airports and borders.
He also said that there would be a higher police presence at carnival processions on Monday.