  The class of Ruddlesden-Popper type (PEA)$_2$PbI$_4$ perovskites comprises
two-dimensional (2D) structures whose optical properties are determined by
excitons with a large binding energy of about 260 meV. It complements the
family of other 2D semiconductor materials by having the band structure typical
for lead halide perovskites, that can be considered as inverted compared to
conventional III-V and II-VI semiconductors. Accordingly, novel spin phenomena
can be expected for them. Spin-flip Raman scattering is used here to measure
the Zeeman splitting of electrons and holes in a magnetic field up to 10 T.
From the recorded data, the electron and hole Land\'e factors ($g$-factors) are
evaluated, their signs are determined, and their anisotropies are measured. The
electron $g$-factor value changes from $+2.11$ out-of-plane to $+2.50$
in-plane, while the hole $g$-factor ranges between $-0.13$ and $-0.51$. The
spin flips of the resident carriers are arranged via their interaction with
photogenerated excitons. Also the double spin-flip process, where a resident
electron and a resident hole interact with the same exciton, is observed
showing a cumulative Raman shift. Dynamic nuclear spin polarization induced by
spin-polarized holes is detected in corresponding changes of the hole Zeeman
splitting. An Overhauser field of the polarized nuclei acting on the holes as
large as $0.6$ T can be achieved.
