I have dismissed an employee due to their medical capability, and they are saying that their notice pay is ‘protected’. What does this mean? We only give the notice we have to under the law, but this employee is signed off long-term sick. Surely we don’t have to pay them?

‘Protected’ notice arises when an employee is dismissed, or resigns, and the following circumstances apply: 

•    They are ready and willing to work but no work is available (eg a lay off situation)
•    They are incapable of work because of sickness or injury
•    They are absent from work because of pregnancy or childbirth, or on adoption parental or paternity leave (exercising their right to family friendly leave)
•    They are absent from work due to holiday.

Where this is the case, and their notice period is the same or not more than a week more than the minimum notice required by the Employment Rights Act 1996 (ie a week for every year of service, to a maximum of 12), then ‘protected notice’ applies. This means that rather than being paid what they normally would be paid for the type of leave they are on during the notice period, such as maternity pay, sick pay, or nothing if there is a lay off situation, they would instead get full pay for the entire notice period. 
In your circumstances, as the employee is off sick, and only paid statutory notice, the protected notice rules apply and they should receive full pay for the duration of their notice period. This can be made up of sick pay and normal pay; there is no requirement to pay their full pay and sick pay on top of that.