A police officer chasing down a kid driving a Corvette
A car chase was when a car or any other type of automobile was in pursuit of another vehicle. A pursuit involving law enforcement was called a police chase.
In 2022, James T. Kirk (from an alternate future) and La'An Noonien-Singh were in a Dodge Challenger Hellcat in pursuit of a van that took something from the scene in which the Lake Ontario Bridge was destroyed by a photonic bomb. Upon pursuing the vehicle, Kirk and Noonien-Singh were in turn pursued by the Toronto City Police, which ended with the police placing Kirk under arrest. Luckily, they were stalled by a woman that was recording them with her cell phone, streaming them live on the internet. Just as they were about to have the woman move along and go through with the arrest, they received a radio call that they were needed at the attack site, and so they released him instead. (SNW: "Tomorrow and Tomorrow and Tomorrow")
In 2024, Seven of Nine and Raffaela Musiker stole an SUV from the parking lot of the Los Angeles Police Department Headquarters. A car chase ensued as the LAPD pursued them in their police cars. (PIC: "Watcher")
In the alternate reality in the mid-2240s, an Iowa State Police officer on a hovercruiser chased down a Corvette convertible that was being driven by Jim Kirk. The high-speed chase ended in a car crash when Kirk intentionally drove it into a quarry, leaping from the vehicle in the nick of time. (Star Trek)
In 2379, while on an away mission on Kolarus III, the away team was in a Starfleet ground vehicle while getting chased by Kolaran ground forces in buggies and an armored car. Data controlled the shuttle via remote control, and Captain Jean-Luc Picard was able to drive the vehicle directly into the rear door of the shuttlecraft. (Star Trek Nemesis)
In 2382, while on Targalus IX, Brad Boimler's away team was kidnapped and loaded into a hover-van by natives that were capitalist extremists. He and Beckett Mariner then commandeered a hover-cruiser to pursue them to a mansion, where they rescued them. (LD: "Shades of Green")
In 2384, a Xindi-Reptilian security team was using their hover speeders to chase down the crew of the USS Protostar in a hover skiff on Teenax. (PRO: "Crossroads")
In 3191, Moll and L'ak were pursued by the crew of the USS Discovery on sand runner motorcycles. (DIS: "Red Directive")