Firm told me I need a work permit

Up until the date when Britain actually leaves the EU (March 2019) – and probably until the end of any transitional period (this will be the end of 2020 assuming all is finalised as planned) – Britons in France remain full EU citizens, with all associated rights, and are entitled to be treated as such.

In other words they cannot be obliged to have a carte de séjour (although it is highly recommended to apply for one as soon as possible so as to consolidate your rights before Brexit) and certainly are not required to hold a work permit to work in France. The latter is a formality which only applies to third country (non-EU) citizens.

If you have evidence that the non-renewal was motivated by the fact you are British you may be able to take action for discrimination. Consider applying to the Défenseur des droits (defenseurdesdroits.fr), a watchdog on discrimination matters who may be able to investigate and advise you further on your rights and options.

One option would be to report the matter to a prud’hommes work tribunal who can look at passing a ruling annulling the employer’s decision and compensating you. It is also possible to make a criminal complaint of discrimination to the police or in writing to the procureur de la République at the tribunal de grande instance.

Note that you can also obtain free legal advice at maisons de justice et du droit. The British consular network is another possible source of help.