We present evidence of non-excimer-based secondary scintillation in gaseous xenon, obtained using both the NEXT-White time projection chamber (TPC) and a dedicated setup. Detailed comparison with first-principle calculations allows us to assign this scintillation mechanism to neutral bremsstrahlung (NBrS), a process that is postulated to exist in xenon that has been largely overlooked. For photon emission below 1000 nm, the NBrS yield increases from about 10-2 photon/e- cm-1 bar-1 at pressure-reduced electric field values of 50 V cm-1 bar-1 to above 3×10-1 photon/e- cm-1 bar-1 at 500 V cm-1 bar-1. Above 1.5 kV cm-1 bar-1, values that are typically employed for electroluminescence, it is estimated that NBrS is present with an intensity around 1 photon/e- cm-1 bar-1, which is about 2 orders of magnitude lower than conventional, excimer-based electroluminescence. Despite being fainter than its excimeric counterpart, our calculations reveal that NBrS causes luminous backgrounds that can interfere, in either gas or liquid phase, with the ability to distinguish and/or to precisely measure low primary-scintillation signals (S1). In particular, we show this to be the case in the "buffer"region, where keeping the electric field below the electroluminescence threshold does not suffice to extinguish secondary scintillation. The electric field leakage in this region should be mitigated to avoid intolerable levels of NBrS emission. Furthermore, we show that this new source of light emission opens up a viable path toward obtaining S2 signals for discrimination purposes in future single-phase liquid TPCs for neutrino and dark matter physics, with estimated yields up to 20-50 photons/e- cm-1.