University Hospitals of Morecambe Bay NHS Foundation Trust (UHMBT), which operates the Royal Lancaster Infirmary, Furness General Hospital in Barrow and Westmorland General Hospital in Kendal, has taken the decision to cancel non-urgent procedures, and is asking people to contact 111 for help and advice rather than attending A&E.
Trust boss Aaron Cummins said a high level of staff sickness, as well as increasing numbers of patients with Covid and in A&E, has led to the decision being made.
But he emphasised that cancer and other urgent operations and appointments will continue to be prioritised.
In a statement, trust chief executive Aaron Cummins said: "We are running at extremely high levels of occupancy and like a number of other hospitals across the country, we are seeing huge rise in demands for our services at the moment.
"Attendances at A&E are rising; there are ongoing challenges in discharging patients who are well enough to leave hospital, and we are seeing an increase in staff sickness - all of which leads to longer waits than we would like for patients to be seen and admitted.
"Because of these pressures, the trust has taken the decision to escalate to its highest level of internal incident which allows us to be able to take additional steps to maintain safe services for our patients and help us cope with the growing pressures.
"Unfortunately, this does mean postponing some non-urgent operations and outpatient appointments to accommodate those patients with the most urgent clinical need.
"Cancer and our other most urgent operations and appointments will continue to be prioritised and those patients whose appointments are being postponed are in the process of being contacted directly by our teams.
"If you are not contacted directly, please continue to turn up for your appointment. Please also contact the number on your appointment letter if you are no longer able to make your appointment so it can be reallocated to another patient.
"We appreciate that this may be disappointing for those patients whose appointments need to be postponed - especially given the ongoing delays due to the pandemic.
"However, patient safety must take priority and we need to take action to make sure that those who are more clinically urgent get the help they need as quickly as possible.
"Our teams continue to work exceptionally hard; and we would like to reassure our patients and the public that in spite of the challenges faced and some changes to non-urgent appointments, essential services remain fully open for anyone who needs them so if you require urgent medical help, please continue to come forward.
"The public can help us manage these periods of demand by ensuring they are seeking help from the most appropriate health services, and only attending A&E for serious accidents and emergencies.
"If you are unwell, visit NHS 111 online for 24/7 advice about the most appropriate care for your needs."
On Twitter, Lancashire's director of public health Dr Sakthi Karunanithi said: "This is to be expected given high number of cases. I know the team at UHMBT will do all they can in keeping us going. We all have a role to play."