Hospital capacity has “regularly” reached over 100 per cent in recent weeks, officials said.
Patients are now facing waits of over ten hours at “overcrowded” Accident and Emergency departments.
The health board has urged patients to seek alternatives to A&E if possible, and to expect long waits as pressure continues to rise.
It comes after the health board banned all but essential hospital visiting, amid rising Covid cases.
The current crisis has been caused by increased numbers of Covid patients, staff self-isolating, and continued staffing shortages, said NHS Lanarkshire deputy director of acute services Russell Coulthard.
He said: “Our three acute sites are full. In fact, we have been working beyond maximum capacity within our hospitals over the past few weeks with capacity regularly over 100 per cent.
“Unfortunately many patients in our emergency department will be waiting over ten hours to be admitted due to the current overwhelming demand which is well in excess of our target of four hours.
“Covid-19 numbers are on the rise with 194 Covid patients across the three acute sites today (Friday). To put that in perspective, numbers had fallen to 66 in the middle of last month but have been increasing ever since and show no sign of levelling off or decreasing.
“The knock-on effect of that is we have staff absences due to Covid and we continue to face significant staffing challenges across health services and our staff are struggling to cope.”
Dr Calvin Lightbody, A&E consultant at University Hospital Hairmyres, said: “We are seeing exceptionally high numbers of people attending our three emergency departments. It is extremely busy and overcrowded but we are working our best to try to get through the huge number of patients.
“If you have an emergency condition such as chest pain, breathing difficulties, severe pain or major bleeding then you should come to A&E as usual and we are very much open for business.”
But Dr Lightbody urged patients who do not have a critical emergency to call NHS 24, or visit their GP or local pharmacy.
"If you do come to A&E, you may face a wait of several hours to be seen,” he said.
"Can I ask you to please be kind and considerate to our staff. We are working extremely hard in very difficult circumstances to do the very best we can for our patients.”
A Scottish Government spokesperson said: “NHS staff continue to face unprecedented pressures as they work to respond to the pandemic whilst continuing to provide vital treatment and optimal patient care.
“Weekly performance continues to be impacted by the direct and indirect impacts of the pandemic such as capacity issues as a result of staff absence and reduced beds due to infection control requirements.
"Hospitals continue to report high numbers of patients presenting who are acutely unwell leading to a longer length of time spent in hospital and impacting on flow. We continue to working closely with our Health Board colleagues, inc Lanarkshire and their partners and are monitoring the situation closely.”