Some unitals on 28 points and their embeddings in projective planes of order 9

We answer three questions posed by F. Piper by exhibiting (i) a unital that is not embeddable in a projective plane, (ii) a unital which is embeddable, and isomorphic with its dual, but not the set of absolute points of a polarity (in fact examples exist in each of the four known projective planes of order 9), (iii) a unital that can be embedded in two different planes.