Let $$u\colon B^n(0,1)\to \mathbb{R}$$ be a subharmonic function in the open unit ball in $\mathbb{R}^n$. The crucial assumption is that $u$ never equals $-\infty$.

Is it true that $|u|$ is bounded on any compact subset of the unit ball?

Remark. (1) $u$ is bounded above on any compact subset of the unit ball since it is upper semi-continuous.

(2) If the answer has the negative answer in the above generality, then I would like to ask it for $u$ being a plurisubharmonic function in the unit ball in $\mathbb{C}^{n/2}=\mathbb{R}^n$.