I have been working on risk assessments and driving at work, can you tell me if it is a requirement for cars used for driving at work to be equipped with a fire extinguisher

No.

Extinguishers are only required where there is a specific requirement under specific legislation, such as foam extinguishers in buses, coaches & minibuses (C&U Regs) and vehicles carrying dangerous goods covered by ADR

Further to AB reply check out  The Regulatory Reform (Fire Safety) Order 2005 article 6 premises exempt and states,

(e) an aircraft, locomotive or rolling stock, trailer or semitrailer used as a means of transport or a vehicle for which a licence is in force under the Vehicle Excise and Registration Act 1994[14] or a vehicle exempted from duty under that Act;

but subject to the legislation AB said.