No. In three dimensions there are three position operators, $\hat{x}_1$, $\hat{x}_2$, and $\hat{x}_3$, or maybe $\hat{x}$, $\hat{y}$, and $\hat{z}$. Each of these is a linear operator in the first, correct sense. Each one maps states in the Hilbert space onto other maps in the Hilbert space and nothing more.
Now, the three distinct position operators are actually closely related to each other, so we often write $\hat{\vec{x}}$ as a shorthand for talking about all of them at once, but they are still three separate operators that map $H\rightarrow H$.