Can the image of a not-bounded "projector" on a normed space be closed?

Let $X$ be a normed space, and $P:X\to X$ be an idempotent linear map, i.e., $P^2=P$. If $P$ is bounded, then $P(X)$ is closed. Does the converse hold? That is, if $P$ is not bounded, must $P(X)$ not be closed? Does the answer change if $X$ is a Banach space?

Take any example of an unbounded operator $T:Y\to Z$ between two normed spaces (they can be Banach spaces if you want). Now consider the operator $P:Y\oplus Z\to Y\oplus Z$ defined by $P(y,z)=(0,z+T(y))$. This operator is idempotent and its image is $Z$, which is closed in $Y\oplus Z$. However, it is not bounded because $T$ was not bounded.