We discuss the possibility that a Born-Infeld condensate coupled to neutrinos can generate both neutrino masses and an effective cosmological constant. In particular, an effective field theory is provided capable of dynamically realizing the neutrino superfluid phase firstly suggested by Ginzburg and Zharkov. In such a case, neutrinos acquire a mass gap inside the Born-Infeld ether forming a long-range Cooper pair. Phenomenological implications of the approach are also discussed. On the other hand, a condensate could be derived from the Born-Infeld theory. In particular, the idea of an invisible Born-Infeld condensate acting as dark energy density was first suggested in Ref. [5]. It is worth mentioning that also the standard visible electrodynamics can be extendedá lá Born-Infeld so that it could contribute, in some sense, to the vacuum energy [6,7].Here, we want to show how an invisible Born-Infeld condensate can be related to the neutrino masses and give rise to the dark energy content of the Universe.The old theory of a non-linear electrodynamics was suggested as a non-linear extension of Maxwell electrodynamics [10]. The main purpose of that model was to solve the insidious self-energy problem of point-like charged particles. This model has a certain attractiveness: it is invariant under Dirac electric-magnetic duality, it predicts the existence of solitonic objects and it has physical propagations without shock waves and dichroism [11]. Even if the original attempts were not successful for electron-self-energy, it turns out to be deeply connected to string theory. The Born-Infeld electrodynamics is the low-energy theory of open strings and it also re-appears as a contribution part of world-volume action of D3-branes as string solitons. So that it is well known, in string theory, how to regularize a Born-Infeld theory in the UV limit. Otherwise problems with quantization and unitarity would severely afflict the proposal. So that, in the contest of D-brane worlds or intersecting D-brane worlds cosmology, we can envisage an invisible electrodynamics confined on a D3-brane and gravitationally interacting in our D-brane world. This picture arXiv:1607.05706v2 [gr-qc]