Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω1 of weight ℵ1, such that the space C(Kω1) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω1 is a continuous image of a Valdivia compact and every separable subspace of C(Kω1) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.