  (ABRIDGED) We present a study of the highly ionized gas (plasma) associated
with damped Lyman-alpha (DLA) systems at z=2.1-3.1. We search for O VI
absorption and corresponding Si IV, C IV, and N V in a Very Large
Telescope/Ultraviolet-Visible Echelle Spectrograph (VLT/UVES) sample of 35 DLA
systems with data covering O VI at S/N>10. We report twelve DLAs (nine
intervening and three at <5000 km/s from the QSO redshift) with detections of O
VI absorption. There are no clear O VI non-detections, so the incidence of O VI
in DLAs is between 34% (12/35) and 100%. Analysis of the line widths together
with photoionization modelling suggests that two phases of DLA plasma exist: a
hot, collisionally ionized phase (seen in broad O VI components), and a warm,
photoionized phase (seen just in narrow C IV and Si IV components). We find
tentative evidence (98% confidence) for correlations between the DLA
metallicity (measured in the neutral gas) and high-ion column density, and
between the DLA metallicity and high-ion line width, as would be expected if
supernova-driven galactic outflows rather than accretion produced the high
ions. Using conservative ionization corrections, we find lower limits to the
total hydrogen column densities in the hot (O VI-bearing) and warm (C
IV-bearing) phases in the range log N(Hot H II) >19.5 to >21.1, and log N(Warm
H II) >19.4 to >20.9. On average, the hot and warm phases thus contain >40% and
>20% of the baryonic mass of the neutral phase in DLAs, respectively. If the
temperature in the O VI phase is ~10^6 K and so f(O VI)=O VI/O<<0.2 the plasma
can make a significant contribution to the metal budget at high redshift.
