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.
