  The main purpose of this paper is to develop some methods to investigate the
Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in
Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic
mappings to Banach spaces, and then we apply these extensions to establish a
sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls
of $\C^n$ and to obtain some sharp boundary Schwarz type lemmas for holomorphic
mappings in Banach spaces. Furthermore, we improve and generalize the classical
Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach
spaces, and present some applications to sharp boundary Schwarz type lemmas for
pluriharmonic mappings in Banach spaces. Additionally, using a relatively
simple method of proof, we prove some sharp Schwarz-Pick type estimates of
pluriharmonic mappings in JB$^*$-triples, and the obtained results provide the
improvements and generalizations of the corresponding results in \cite{CH20}.
