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We complete the classification, up to isomorphism, of the spaces of compactoperators on C([1, gamma], l_p) spaces, 1<p< infinity. In order to do this, weclassify, up to isomorphism, the spaces of compact operators {\mathcal K}(E,F), where E= C([1, lambda], l_p) and F=C([1,xi], l_q) for arbitrary ordinalslambda and xi and 1< p \leq q< infinity.  More precisely, we prove that it is relatively consistent with ZFC that forany infinite ordinals lambda, mu, xi and eta the following statements areequivalent:  (a) {\mathcal K}(C([1, lambda], l_p), C([1, xi], l_q)) is isomorphic to{\mathcal K}(C([1, mu], l_p), C([1, eta], l_q)) .  (b) lambda and mu have the same cardinality and C([1,xi]) is isomorphic toC([1, eta]) or there exists an uncountable regular ordinal alpha and 1 \leq m,n < omega such that C([1, xi]) is isomorphic to C([1, alpha m]) and C([1,eta])is isomorphic to C([1, alpha n]).  Moreover, in ZFC, if lambda and mu are finite ordinals and xi and eta areinfinite ordinals then the statements (a) and (b') are equivalent.  (b') C([1,xi]) is isomorphic to C([1, eta]) or there exists an uncountableregular ordinal alpha and 1 \leq m, n \leq omega such that C([1, xi]) isisomorphic to C([1, alpha m]) and C([1,eta]) is isomorphic to C([1, alpha n]).

A complete classification of the spaces of compact operators on C([1,alpha], l_p) spaces, 1