  We show that there are no automorphic Banach spaces of the form C(K) with K
continuous image of Valdivia compact except the spaces c0(I). Nevertheless,
when K is an Eberlein compact of finite height such that C(K) is not isomorphic
to c0(I), all isomorphism between subspaces of C(K) of size less than
aleph_omega extend to automorphisms of C(K).
