Although Lanzarote is a long way from Spain, it is Spanish territory and therefore all passport and residency rules are the same as on the Spanish mainland.
European Union citizens can travel to Lanzarote with national ID cards alone. If your country does not issue identity cards, then a valid Passport is required. UK visitor passports are not acceptable. All non EU nationals require a full valid Passport.
Carry your ID card or Passport with you at all times. The police may ask to see it. As a backup, make photocopies of all your valuable documents and leave copies in your hotel accommodation as well as at home.
Nationals of the EU as well as Norway and Iceland do not require a Visa for a stay of up to 90 days. If you wish to stay longer than 90 days you will need to apply for a Residency Permit during the first month of your stay.