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

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