  Results on $\Lambda$ hyperon production are reported for collisions of p(3.5
GeV) + Nb, studied with the High Acceptance Di-Electron Spectrometer (HADES) at
SIS18 at GSI Helmholtzzentrum for Heavy-Ion Research, Darmstadt. The transverse
mass distributions in rapidity bins are well described by Boltzmann shapes with
a maximum inverse slope parameter of about $90\,$MeV at a rapidity of $y=1.0$,
i.e. slightly below the center-of-mass rapidity for nucleon-nucleon collisions,
$y_{cm}=1.12$. The rapidity density decreases monotonically with increasing
rapidity within a rapidity window ranging from 0.3 to 1.3. The $\Lambda$
phase-space distribution is compared with results of other experiments and with
predictions of two transport approaches which are available publicly. None of
the present versions of the employed models is able to fully reproduce the
experimental distributions, i.e. in absolute yield and in shape. Presumably,
this finding results from an insufficient modelling in the transport models of
the elementary processes being relevant for $\Lambda$ production, rescattering
and absorption. The present high-statistics data allow for a genuine
two-dimensional investigation as a function of phase space of the
self-analyzing $\Lambda$ polarization in the weak decay $\Lambda\rightarrow p
\pi^-$. Finite negative values of the polarization in the order of $5-20\,\%$
are observed over the entire phase space studied. The absolute value of the
polarization increases almost linearly with increasing transverse momentum for
$p_t>300\,$MeV/c and increases with decreasing rapidity for $y < 0.8$.
