We prove the following quasi-dichotomy involving the Banach spaces C(α,X) of all X-valued continuous functions defined on the interval [0,α] of ordinals and endowed with the supremum norm.
Suppose that X and Y are arbitrary Banach spaces of finite cotype. Then at least one of the following statements is true.
(1) There exists a finite ordinal n such that either C(n,X) contains a copy of Y, or C(n,Y) contains a copy of X.
(2) For any infinite countable ordinals α, β, ξ, η, the following are equivalent:
(a) C(α,X) ⊕ C(ξ,Y) is isomorphic to C(β,X) ⊕ C(η,Y).
(b) C(α) is isomorphic to C(β), and C(ξ) is isomorphic to C(η).
This result is optimal in the sense that it cannot be extended to uncountable ordinals.