You want to take the category $\text{Ban}_1$ of Banach spaces and short maps (linear maps of operator norm $\le 1$). The unit ball functor $U : \text{Ban}_1 \to \text{Set}$ is represented by $\mathbb{C}$, and its left adjoint sends a set $S$ to the coproduct of $S$ copies of $\mathbb{C}$, which turns out to be $\ell^1(S)$. This says that we have a natural bijection
$$\text{Hom}_{\text{Ban}_1}(\ell^1(S), B) \cong \text{Hom}_{\text{Set}}(S, U(B))$$
which says that a map from a set $S$ to the unit ball $U(B)$ of a Banach space extends uniquely and freely to a short map $\ell^1(S) \to B$, by "linearity."
Intuitively speaking this says that $\ell^1(S)$ is obtained from $S$ by requiring that each element of $S$ have norm $1$ (so that it's in the unit ball and can map shortly to any other element of any other unit ball) and then asking that a linear combination $\sum c_s s$ have the largest possible norm compatible with this (so that it can map shortly to any other such linear combination in any other Banach space). We have $ \| \sum c_s s \| \le \sum |c_s|$ by the triangle inequality and the $\ell^1$ norm is the equality case of this.
This construction generalizes to the construction of the coproduct in $\text{Ban}_1$, which looks like this: if $B_i$ is a collection of Banach spaces, their coproduct in $\text{Ban}_1$ is the completion of the vector space direct sum $\bigoplus_i B_i$ with respect to the "$\ell^1$ norm" $\sum_i \| b_i \|_{B_i}$.
Apologies for the self-promotion, but I go into a bit more detail about categorical properties of $\text{Ban}_1$ (e.g. it is complete, cocomplete, and closed symmetric monoidal) in my blog post Banach spaces (and Lawvere metrics, and closed categories). In particular I attempt to motivate the use of short maps. Note that if we only work with bounded linear maps then we can't hope to recover a Banach space up to isometry via a universal property, whereas the isomorphisms in $\text{Ban}_1$ are isometric. On the other hand the categorical language is still capable of talking about bounded maps, via the closed structure.