  Context: Lithium is a fragile element, which is easily destroyed in the
stellar interior. The existence of lithium-rich giants still represents a
challenge for stellar evolution models. Aims: We have collected a large
database of high-resolution stellar spectra of 824 candidate thick-disk giants
having 2\,MASS photometry and proper motions measured by the Southern
Proper-Motion Program (SPM). In order to investigate the nature of Li-rich
giants, we searched this database for giants presenting a strong Li\,I
resonance line. Methods: We performed a chemical abundance analysis on the
selected stars with the MOOG code along with proper ATLAS-9 model atmospheres.
The iron content and atmospheric parameters were fixed by using the equivalent
width of a sample of Fe lines. We also derive abundances for C, N, and O and
measure or derive lower limits on the $^{12}$C/$^{13}$C isotopic ratios, which
is a sensible diagnostic of the stars evolutionary status. Results: We detected
five stars with a lithium abundance higher than 1.5, i.e. Li-rich according to
the current definition. One of them (SPM-313132) has A(Li)$>$3.3 and, because
of this, belongs to the group of the rare super Li-rich giants. Its kinematics
makes it a likely thin-disk member and its atmospheric parameters are
compatible with it being a 4\,M$_\odot$ star either on the red giant branch
(RGB) or the early asymptotic giant branch. This object is the first super
Li-rich giant detected at this phase. The other four are likely low-mass
thick-disk stars evolved past the RGB luminosity bump, as determined from their
metallicities and atmospheric parameters. The most evolved of them lies close
to the RGB-tip. It has A(Li)$>$2.7 and a low $^{12}$C/$^{13}$C isotopic ratio,
close to the cool bottom processing predictions.
