Can a fighter ever be purely defensive?

Has there ever been a fighter that could only perform defensive work?

Probably not in a strict sense, because even lightly armed fighters can usually turn their weapons against air, ground, or naval targets.

A fighter may not remain purely defensive once it carries a projectile weapon. First World War aircraft such as the Sopwith Camel primarily fought other aircraft but also used machine guns and small bombs against trenches and other targets. The same weapon that intercepts an attacker can often be redirected against a surface objective. That flexibility supports the definition of carrier craft as contributors to both attack and defense rather than as separate single-purpose categories. Early Spitfires also used their machine guns against torpedo boats.