Do I need a written tenancy agreement?

Not legally, but letting without one is a bad idea and makes every later dispute harder.

A tenancy exists as soon as someone pays rent for exclusive occupation, written down or not. What you lose without a document is any record of what was agreed about rent, deposit, bills, pets or access.

Some things must be in writing whatever the agreement says. The deposit prescribed information, the How to Rent guide, the EPC and the safety certificates all have to be given to the tenant and evidenced.