  The Fock-Bargmann-Hartogs domain $D_{n, m}(\mu)$ is defined by $$ D_{n,
m}(\mu):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w
\Vert^2<e^{-\mu\Vert z \Vert^2}\},$$ where $\mu>0.$ The Fock-Bargmann-Hartogs
domain $D_{n, m}(\mu)$ is an unbounded strongly pseudoconvex domain with smooth
real-analytic boundary. In this paper, we first compute the weighted Bergman
kernel of $D_{n, m}(\mu)$ with respect to the weight $(-\rho)^{\alpha}$, where
$\rho(z,w):=\|w\|^2-e^{-\mu \|z\|^2}$ is a defining function for $D_{n,
m}(\mu)$ and $\alpha>-1$. Then, for $p\in [1,\infty),$ we show that the
corresponding weighted Bergman projection $P_{D_{n, m}(\mu), (-\rho)^{\alpha}}$
is unbounded on $L^p(D_{n, m}(\mu), (-\rho)^{\alpha})$, except for the trivial
case $p=2$. In particular, this paper gives an example of an unbounded strongly
pseudoconvex domain whose ordinary Bergman projection is $L^p$ irregular when
$p\in [1,\infty)\setminus\{2\}$. This result turns out to be completely
different from the well-known positive $L^p$ regularity result on bounded
strongly pseudoconvex domain.
