Tenancy at sufferance - or holdover tenancy - is when a tenant continues paying the lease even after it has expired. If the landlord accepts payment for the lease, the tenant may legally remain on the estate. However, if the tenant fails to comply with the terms of the original lease, the landlord can initiate eviction procedures.
In some cases the lease terms are reset if the landlord accepts payment. In other cases it simply prolongs the lease for another month. This depends on local legislation or can regulated in the lease itself.
If a landlord wants to remove the tenant from the estate, they have to decline accepting payment and initiate eviction procedures - in other words: to treat them like a trespasser. The court proceedings that follow are referred to as “holdover proceedings”. These are eviction cases that are not based on missed rental payments. They can involve trying to remove squatters from the property who never had a lease to begin with or if the tenant caused a public nuisance on the estate, for example.
If both parties want to avoid a tenancy at sufferance, the landlord could offer a new lease. Alternatively, even a written and signed one-sentence agreement stating that the original duration of the lease is extended by a certain period of time would clarify matters.
It is important to keep in mind that even with eviction proceedings underway the landlord could still be forced to accept tenancy at sufferance if the tenant continues to pay the lease and that these eviction proceedings can take several months in court.