I'm reading about quantum anomalies in QFT and all the examples seem to arise in gauge theories. Is it true that theories without a local gauge invariance don't have quantum anomalies? I can't think of examples of symmetry violation in any non-gauge theories, particularly the free theories. Then again, there might be a well-known example I'm not aware of!
There are things called sigma model anomalies, see papers listed in a sample inspire database query here.
Here, the anomaly is associated to the general coordinate invariance in the target space of the non-linear sigma model: the fields take values in a nontrivial manifold (and its associated vector bundles), rather than vector spaces. Classically, the action is independent of the coordinates used to describe the target manifold. But this independence can be lost via quantization.
A more general definition of anomaly: A QFT that has no UV completion in the same dimension is anomalous.
In other words, a QFT that has no well defined short distance regularization in the same dimension is anomalous.
Example: A 1+1D QFT with only one right moving fermion mode is anomalous.