Can an employee claim unfair dismissal on expiry of a fixed-term contract of employment?

Yes, the expiry of a fixed term is classed as a dismissal in law, meaning that, under the Employment Rights Act 1996, the employee could claim unfair dismissal and (potentially) redundancy pay, provided that they have two years' service. To defend an unfair dismissal claim, the employer would have to show that it had a fair reason for dismissing the employee (for example redundancy or "some other substantial reason") and that it acted fairly and reasonably in dismissing the employee for that reason.