  The purpose of this paper is to introduce and study $\lambda$-infinitesimal
BiHom-bialgebras (abbr. $\l$-infBH-bialgebra) and some related structures. They
can be seen as an extension of $\l$-infinitesimal bialgebras considered by
Ebrahimi-Fard, including Joni and Rota's infinitesimal bialgebras as well as
Loday and Ronco's infinitesimal bialgebras, and including also infinitesimal
BiHom-bialgebras introduced by Liu, Makhlouf, Menini, Panaite. In this paper,
we provide various relevant constructions and new concepts. Two ways are
provided for a unitary (resp. counitary) algebra (coalgebra) to be a
$\l$-infBH-bialgebra and the notion of $\l$-infBH-Hopf module is introduced and
discussed. It is proved, in connexion with nonhomogeneous (co)associative
BiHom-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.
