Rigid first-order structure

A first-order structure is rigid when its identity map is its only automorphism. Every finite linear order is rigid, whereas $(\mathbb{Z},<)$ is nonrigid because every translation is an order automorphism.

If a theory has arbitrarily large finite nonrigid models, expand its language by a unary function $f$, axiomatize that $f$ is an automorphism, require $f$ to move some element, and add sentences demanding at least $n$ elements for every $n$. Every finite fragment has a sufficiently large finite nonrigid model, so compactness supplies an infinite nonrigid model.