We present a number of results concerning infinite-exponent partition relations on linear orders of the form $\langle {}^α2,<_{\text{lex}}\rangle$ for $α$ an ordinal, generalising the setting of the real line, working throughout in ZF without the Axiom of Choice. As a particular consequence of our results, we obtain a full classification of the relation $\langle {}^α2,<_{\text{lex}}\rangle \rightarrow (τ)^τ$ for $τ$ countable.