More than one person - each tenant is a Responsible Person under the legislation as an employer having control over part of the premises and/or any other person having control of the premises (under a lease). So each tenant is required by law to carry out and record (regardless of staff or other numbers) a fire risk assessment and within their demise to provide what general fire precautions are required by that assessment, which would include extinguishers, signs, emergency lighting, etc.
The landlord is only responsible for any areas where there is no other Resonsible Person such as a communal stairway, plant room, etc. or they have control over a common feature (e.g. a common building wide fire alarm).
So a three storey building with, let's say, three tenants and a common access stair, lift motor room, electrical intake and toilets would have 4 Fire Risk Assessments, each RP providing whatever is needed in their area. There is, in addition, a requirement for Responsible Persons to share information & coordinate fire safety between them where they share parts of one building.
https://www.gov.uk/government/publications/people-with-duties-under-fire-safety-laws