  [ABRIDGED] We determine Li abundances for a well-studied sample of 714 F and
G dwarf, turn-off, and subgiant stars in the solar neighbourhood. The analysis
is based on line synthesis of the Li line at 6707 {\AA} in high-resolution and
high signal-to-noise ratio echelle spectra, obtained with the MIKE, FEROS,
SOFIN, UVES, and FIES spectrographs. The presented Li abundances are corrected
for non-LTE effects. Out of the sample of 714 stars we are able to determine Li
abundances for 420 stars and upper limits on the Li abundance for another 121
stars. 18 of the stars with well-determined Li abundances are listed as
exoplanet host stars. Our main finding is that there are no signatures of Li
production in the thick disk, but the Li abundance for stars of the same
effective temperature is independent of metallicity for stars that can be
associated with the Galactic thick disk. Significant Li production is however
seen in the thin disk, with a steady increase towards super-solar
metallicities. At the highest metallicities, however, around [Fe/H]~ +0.3, we
tentatively confirm the recent discovery that the Li abundances level out. We
hence contradict the recent finding in other studies that found that Li is also
produced in the thick disk. This is likely due to the chemically defined
selection criteria those studies used to define their thick disk samples. Age
criteria that we use here, produce a thick disk stellar sample that is much
less contaminated by thin disk stars, and hence more reliable abundance trends.
[ABRIDGED] A conclusion that can be drawn is that no significant Li production,
relative to the primordial abundance, took place during the first few billion
years of the Milky Way, an era coinciding with the formation and evolution of
the thick disk. [ABRIDGED]
