As a business owner or employer, the law requires that you provide each employee with an itemized statement of their pay, on or before pay day. As long as you have done this, there’s no legal requirement to provide copies of past payslips.
You are, of course, required to provide the employee with a summary of their year’s pay in the form of their P60 or P45. However, it’s up to your employee to retain their individual payslips for as long as they may require them.
Some employers now provide an electronic payslip system. This enables employees to view and print off past payslips whenever they like, eliminating the problems of lost payslips.