  In this paper, we study a multidimensional backward stochastic differential
equation (BSDE) with an additional rough drift (rough BSDE), and give the
existence and uniqueness of the adapted solution, either when the terminal
value and the geometric rough path are small, or when each component of the
rough drift only depends on the corresponding component of the first unknown
variable (but dropped is the one-dimensional assumption of Diehl and Friz [Ann.
Probab. 40 (2012), 1715-1758]). We also introduce a new notion of the $p$-rough
stochastic integral for $p \in \left[2, 3\right)$, and then succeed in giving
-- through a fixed-point argument -- a general existence and uniqueness result
on a multidimensional rough BSDE with a general square-integrable terminal
value, allowing the rough drift to be random and time-varying but having to be
linear; furthermore, we connect it to a system of rough partial differential
equations.
