Lanzarote is (as part of Spain) within the EU and so, for the moment, is the UK. EU laws require all member countries to admit all EU nationals without restrictions as long as they can prove such status. So the Spanish authorities MUST allow you into their country as long as you've got a valid British (or other EU) passport. Since you don't have to leave if you don't want to (i.e. you're free to live in, and work in, Spain indefinitely) they can't impose any restrictions about the validity of your passport.
i.e. you could turn up at one minute to midnight on the very last day of your passport's validity and (even though it wouldn't be valid for your return journey) they must still grant you admission. (However the Foreign & Commonwealth Office quite reasonably states that you should ensure that your passport is valid until the end of your proposed stay, so that you can travel home without any difficulties!)
Only a few (non-EU) countries have the '6-month validity' rule which travel agencies often quote (simply because they can't be bothered to train their staff properly about the rules for individual countries) and even some of those, such as the USA, waive that requirement for British tourists.
You can check on the current entry requirements (for holders of British passports) for
any country here:
https://www.gov.uk/foreign-travel-advice