I bought a car in January 2020, that developed a fatal fault after 57 miles.
The dealer is refusing to deal with me or accept the vehicle back.
I see from Companies House that the same company is showing as dissolved in September 2019.
My question is, is he breaking the law?
I can't give a specific answer to your question (as I would need to know much more about the cirumstances) but the following points may be of use: (i) if the company was dissolved before you bought the car, you could not have bought the car from the company, as it did not legally exist at the time, and you must have bought it from some other person, natural or legal - perhaps the individual in question acting as a sole trader; (ii) trading using the same name as a dissolved company is not in itself unlawful, although using "Limited" or "Ltd" for a non-company is (see Part 5 and Sch 2 of http://www.legislation.gov.uk/uksi/2015/17); and (iii) for your rights in relation to second hand car purchases from a dealer, see: https://www.which.co.uk/consumer-rights/l/new-and-used-cars.