  In this paper, we prove the existence of a mild $L^p$-solution for the
backward stochastic evolution inclusion (BSEI for short) of the form
\begin{align*}%\label{BSDI3}
  \begin{cases}
  dY_t+AY_tdt\in G(t,Y_t,Z_t)dt+Z_tdW_t,\quad t\in [0,T]
  Y_T =\xi,
  \end{cases} \end{align*} where $W=(W_t)_{t\in [0,T]}$ is a standard Brownian
motion, $A$ is the generator of a $C_0$-semigroup on a UMD Banach space $E$,
$\xi$ is a terminal condition from $L^p(\Omega,\mathscr{F}_T;E)$, with $p>1$
and $G$ is a set-valued function satisfying some suitable conditions.
  The case when the processes with values in spaces that have martingale type
$2$, has been also studied.
