# Does there exist an infinite dimensional compact operator on any infinite dimensional Banach space?

Let $X$ be an infinite dimensional Banach space. Does there always exist a compact linear operator on $X$ with infinite dimensional range? I can see that the answer is yes if $X$ is a Hilbert space, as one can take an orthonormal basis containing a countable orthonormal set $\{e_n\}_{n=1}^\infty$, and define a linear operator $T$ so that $T(e_n)=\frac1n e_n$, and zero on the other basis vectors. But I don't see how to construct one on a general Banach space. Any help would be greatly appreciated!