In the general context of scalar-tensor theories, we consider a model in which a scalar field coupled to the Ricci scalar in the gravitational sector of the Lagrangian, is also playing the role of an "Extended Quintessence" field, dominating the energy content of the Universe at the present time. In this framework, we study the linear evolution of the perturbations in the Quintessence energy density, showing that a new phenomenon, named here "gravitational dragging", can enhance the scalar field density perturbations as much as they reach the non-linear regime. The possibility of dark energy clumps formation is thus discussed.1. The growth of quintessence linear perturbations.The recent growing evidences for a vacuum energy component in the Universe [1,2] have lead to the development of a number of models, most of which tend to alleviate the theoretical difficulties of the well known "cosmological constant problem" (see [3] for a review). All these models are replacing the cosmological constant with a "dynamical vacuum energy" provided by an evolving scalar field, often named "Quintessence". This field should provide the present cosmic accelerating phase through its negative equation of state. Such "dark energy" component has also been modeled in the general framework of scalar-tensor theories of Gravity: in these "Extended Quintessence" (EQ) scenarios [4]-[11], the Quintessence scalar field is directly coupled to the Ricci scalar R in the Lagrangian of the theory: the canonical Ricci scalar term κ −1 R, with κ = 8πG * , is replaced by F (φ)R, where F (φ) is a function of the quintessence field φ. Here, G * is a "bare" gravitational constant, generally different from the Newton's constant G as it is measured by Cavendish-type experiments [12,13]. In EQ models, a scalar field has thus a double role: at any epoch, its value determines the effective gravitational constant, and the contribution in dark energy density. Of course, such a coupling is not arbitrary: it is instead subject to a number of constraints, mainly arising from the bounds on the time variations of the constants of nature, which must be taken into account. EQ scenarios have been analyzed from different points of view, and there are robust predictions about their possible imprints on the power spectrum of CMB anisotropies [4,5]. However, the important role of Dark Energy in modern cosmology opens the question of its relation with the other main dark component currently under investigation, the Dark Matter. One major point with Quintessence scenarios is whether a Dark Energy component could affect, in some way, the formation of Dark Matter clumps. In particular, one fundamental question is: can the Dark Energy itself form clumps on certain scales? If we restrict ourselves to minimally-coupled scalar fields, we have to deal with a very relativistic component: indeed, the sound speed of such component turns out to be identical to that of radiation [14]. As a consequence, an "ordinary" (i.e., minimally coupled) Quintessence component becomes homogeneou...