  To alleviate the discrepancy between the prediction of the primordial lithium
abundance in the universe and the abundances observed in Pop II dwarfs and
subgiant stars, it has been suggested that the stars observable today have
undergone photospheric depletion of Li. To constrain the nature of such
depletion, we conduct a homogeneous analysis of a very large sample of stars in
the metal-poor globular cluster NGC6397, covering well all evolutionary phases
from below the main sequence turn-off to high up the red giant branch. Non-LTE
Li abundances or abundance upper limits are obtained for all stars, and for a
size-able sub-set of the targets also Na abundances are obtained. The sodium
abundances are used to distinguish stars formed out of pristine material from
stars formed out of material affected by pollution from a previous generation
of more massive stars. The dwarfs, turn-off, and early subgiant stars form a
thin abundance plateau, disrupted in the middle of the subgiant branch by the
lithium dilution caused by the first dredge-up. A second steep abundance drop
is seen at the red giant branch bump. The turn-off stars are more lithium-poor,
by up to 0.1 dex, than subgiants that have not yet undergone dredge-up. In
addition, hotter dwarfs are slightly more lithium-poor than cooler dwarfs,
which may be a signature of the so-called Li dip in the cluster, commonly seen
among PopI stars. A considerably large spread in Na abundance confirms that
NGC6397 has suffered from intra-cluster pollution in its infancy and a limited
number of Na-enhanced and Li-deficient stars strongly contribute to form a
significant anti-correlation between the abundances of Na and Li. The inferred
Li abundances are compared to predictions from stellar structure models
including atomic diffusion and ad-hoc turbulence below the convection zone.
