Cardinality of dense set related to cardinality of discrete set

Question

Let $X$ be a Banach space. Let $\eta$ be the density character of $X$ (the least possible cardinality of a dense set). Does there exist $A\subset X$, which is closed and such that $\text{card} A=\eta$ and $A$ does not have limit points? ($A$ is a discrete set). If this is true, how can we relax the assumptions? It it is false as stated, can we say when it is true? Also, what can we say about the relation between the density character and the cardinality of the maximal discrete subset?

Answer 1

In this answer (in particular items 5 and 7 are of interest) I show that the minimal cardinality of a discrete subset and the density character indeed are the same for all metric spaces.

In particular this holds for Banach spaces. (It's well known that any infinite Hausdorff space at least has one countably-infinite discrete subspace)