In this paper, we introduce a new property of Banach spaces called $wMB$-property of order $p$ ($1 \leq p < \infty$). A necessary and sufficient condition for a Banach space to have the $wMB$-property of order $p$ is given. We study $p$-convergent operators and weakly-$p$-$L$-sets. Banach spaces with the $wMB$-property of order $p$ are characterized. Also, the Dunford-Pettis property of order $p$ and $DP^*$-property of order $p$ are studied in Banach spaces. Finally we show the relation between Pelczynski's property $(V)$ and $wMB$-property of order $p$.