Absences have more than doubled since the last week in December, to an average of 7,174 in the week to Tuesday.
The Royal College of Nursing said staff are exhausted and worn down, and called on the Scottish Government to do more to support nurses.
“The increase in Covid-19 related absence for NHS and care home staff is a significant concern,” said interim director Colin Poolman.
"Nursing staff across acute, community and social care services are telling us that they are seeing the impact on their ability to provide safe care for patients and residents, and on their, and their colleagues, wellbeing.”
An average of 3,570 nursing and midwifery absences were recorded each day last week, along with 156 in medical and dental.
NHS Lanarkshire has announced that GPs will enter a “managed suspension of services” for a month to cope with staffing shortages and high demand.
Practices will focus on only the most urgent care, including suspected Covid-19 or cancer.
Patients with less urgent issues have been asked to visit the NHS Lanarkshire website for further advice, or to phone NHS 111 or visit a pharmacy.
The changes came into effect on Tuesday and will continue for four weeks.
It comes after NHS Lanarkshire declared it was at critical occupancy levels in October. Health boards across the country have continued to struggle under pressure since then.
Dr Linda Findlay, Medical Director of South Lanarkshire Health and Social Care Partnership, said: “GPs have been open throughout the pandemic and have been busier than ever. They have changed the way they provide care to help protect their patients, maintain safe infection control measures and minimise physical contact.
“This change to GP practices will help us deal with the continuing challenges in community services and help reduce the pressure across the whole system, including our acute hospitals which are under sustained pressure.”
Dr Keith McIntyre, Chair of Lanarkshire GP Sub-Committee, said: “We are asking Lanarkshire residents for their continued help and support during this challenging time. There are a number of alternatives where people can turn to for health care, which will allow GP practices to focus on the most urgent of cases.
“The services provided under this change will vary between practices depending on their individual circumstances. However, GPs will continue to see patients in-person as and when it is appropriate. Practices have infection control processes in place, including physical distancing.
“People should continue to contact their GP practice for urgent issues, such as if they think they have symptoms of cancer.”
Scottish Labour health spokesperson Jackie Baillie said the step was indicative of an NHS crisis.
"Not long ago escalating to risk level black was almost unheard of, but now not even that is enough,” she said.
“Staff are working tirelessly to do right by patients, but services have been pushed past breaking point.”