Is there a series that's unconditionally convergent but not absolutely convergent?

The two properties are equivalent in R and in finitely-dimensional spaces.

Try \(\displaystyle X=\ell_2\) and the series \(\displaystyle x_n=\frac 1{\sqrt{n}}e_n\), where \(\displaystyle e_n\) denotes the sequence which has one on n'th place and all other terms are zeros.

If I remember correctly, this is precisely the most basic example given in the book

Kadets, Kadets: Series in Banach spaces: conditional and unconditional convergence

which is devoted basically to this topic.

However, many texts on functional analysis mention the relationship between various modes of convergence in Banach spaces.

I remember I have seen this in Wojtaszczyk's Banach spaces for analysts (with proofs) and in Megginson's Intorduction to Banach spaces theory (as an exercise - if I remember correctly).