  We introduce a new quantized enveloping superalgebra
$\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra
${\mathfrak{p}}_n$ of type $P$. The superalgebra
$\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra
structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We
also introduce the periplectic $q$-Brauer algebra and prove that it is the
centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on
${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new
periplectic $q$-Schur superalgebra.
