Let $X$ be Banach, $T: X\rightarrow X $ compact injective. Then $\#X \leq \#\text{ran}T \leq \#\mathbb{R}$ because compact operators have separable range. Thus, no compact injective operator $X\rightarrow X$ exists if $X$ has cardinality that is greater than the continuum.
Do you have another proof of the fact that there doesn't always exist $T: X\rightarrow X$ compact injective, that doesn't use cardinality?
References: Cardinality of separable metric spaces and Cardinality approach
