It comes after A&E departments across Scotland recorded their worst waiting times performance on record this week.
The lowest ever percentage of people were admitted, transferred or discharged within the target time of four hours, while more people than ever waited longer than four, eight and 12 hours.
NHS Ayrshire and Arran said services are under pressure due to staff absences, high demand, delayed discharges and patients staying longer in hospital.
Leaders asked members of the public to seek alternative care where possible.
It comes after several other health boards, including NHS Greater Glasgow and Clyde and NHS Lanarkshire, renewed calls for people to consider care options other than A&E.
"While most patients coming through our Emergency Departments are ill and need to be there, we know that some people who attend our EDs would receive more appropriate and quicker treatment elsewhere,” said Joanne Edwards, Director of Acute Services at NHS Ayrshire and Arran.
“So, we are asking members of the public to stop, think, and ask ‘Is it an emergency?’ If it is an emergency, come straight to the Emergency Department or call 999.
“That way we can ensure that our Emergency Departments are there for those who need it most – those with serious illnesses and accidents such as suspected stroke, heart attack, head injuries, fractures or wounds that need stitches.
“If it’s not an emergency, call NHS 24 on 111 and you will be assessed over the telephone and referred to the right healthcare professional."