  This paper is devoted to tangent martingales in Banach spaces. We provide the
definition of tangency through local characteristics, basic $L^p$- and
$\phi$-estimates, a precise construction of a decoupled tangent martingale, new
estimates for vector-valued stochastic integrals, and several other claims
concerning tangent martingales and local characteristics in infinite
dimensions. This work extends various real-valued and vector-valued results in
this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg,
Kwapie\'{n}, McConnell, and Woyczy\'{n}ski. The vast majority of the assertions
presented in the paper is done under the sufficient and necessary UMD
assumption on the corresponding Banach space.
