  The aim of the paper is to prove the existence and uniqueness of the
$L^{p}$--variational solution, with $p>1,$ of the following multivalued
backward stochastic differential equation with $p$--integrable data:
\begin{equation*} \left\{ \begin{array}[c]{l}
-dY_{t}+\partial_{y}\Psi(t,Y_{t})dQ_{t}\ni
H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<\tau,\\[0.1cm] Y_{\tau}=\eta,
\end{array} \right. \end{equation*} where $\tau$ is a stopping time, $Q$ is a
progresivelly measurable increasing continuous stochastic process and
$\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous
function $y\mapsto\Psi(t,y).$
