Plurisubharmonic domination in Banach spaces

We show that if $X$ is a Banach space with a Schauder basis, $\Omega\subset X$ is a pseudoconvex open subset, and $u:\,\Omega\to(-\infty,\infty)$ is a locally bounded function, then there are a Banach space $Z$ and a holomorphic function $h:\,\Omega\to Z$ with $u(x)<\|h(x)\|$ for all $x\in\Omega$.