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The purpose of this paper is to introduce and study $\lambda$-infinitesimalBiHom-bialgebras (abbr. $\l$-infBH-bialgebra) and some related structures. Theycan be seen as an extension of $\l$-infinitesimal bialgebras considered byEbrahimi-Fard, including Joni and Rota's infinitesimal bialgebras as well asLoday and Ronco's infinitesimal bialgebras, and including also infinitesimalBiHom-bialgebras introduced by Liu, Makhlouf, Menini, Panaite. In this paper,we provide various relevant constructions and new concepts. Two ways areprovided for a unitary (resp. counitary) algebra (coalgebra) to be a$\l$-infBH-bialgebra and the notion of $\l$-infBH-Hopf module is introduced anddiscussed. It is proved, in connexion with nonhomogeneous (co)associativeBiHom-Yang-Baxter equation, that every (left BiHom-)module (resp. comodule)over a (anti-)quasitriangular (resp. (anti-)coquasitriangular)$\l$-infBH-bialgebra carries a structure of $\l$-infBH-Hopf module. Moreover,two approaches to construct BiHom-pre-Lie (co)algebras from$\l$-infBH-bialgebras are presented.

Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and properties