You can claim state pension at pension age whether or not you go on working. If you put off claiming state pension, you may be able to earn extra state pension (see Chapter 46(4)).

If you work after pension age, you will not have to pay national insurance contributions. You will need to give your employer proof of your age (such as a passport or birth certificate).
SSCBA , S.6(3)

If you are employed, are earning £123 a week or more, and have been sick for four or more days in a row, claim statutory sick pay from your employer. There is no age limit.