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In this article, without requiring solidness of the underlying cone, a kind of new convergence for sequences in cone bb-metric spaces over Banach algebras and a new kind of completeness for such spaces, namely, wrtn-completeness, are introduced. Under the condition that the cone bb-metric spaces are wrtn-complete and the underlying cones are normal, we establish a common fixed point theorem of contractive conditions with vector-valued coefficients in the non-solid cone bb-metric spaces over Banach algebras, where the coefficients s≥1s\ge 1. As consequences, we obtain a number of fixed point theorems of contractions with vector-valued coefficients, especially the versions of Banach contraction principle, Kannan’s and Chatterjea’s fixed point theorems in non-solid cone bb-metric spaces over Banach algebras. Moreover, some valid examples are presented to support our main results.