Chains of length $2^kappa$ in ${cal P}(kappa)$

It is a fact that continues to boggle my mind: There is a set ${cal C}subseteq {cal P}(omega)$ such that $|{cal C}|=frak{c}=2^{aleph_0}$ and for all $A,Bin{cal C}$ we have $Asubseteq B$ or $Bsubseteq A$ for all $A,Bin {cal C}$. (In that case we call ${cal C}$ a chain.)

The proof is remarkably easy: pick any bijection $varphi:omegatomathbb{Q}$ and consider the collection ${cal D}$ of all down-sets of $mathbb{Q}$. This collection forms the Dedekind cuts which can be used to construct $mathbb{R}$, which has cardinality $frak{c}$. Now ${varphi^{-1}(D):Din{cal D}}$ is a chain in ${cal P}(omega)$ of cardinality $frak{c}$.

Now you would think that you could carry the construction of $mathbb{Q}$ and Dedekind cuts to cardinals larger than $omega$ — but I was not able to.

Question. If $kappa$ is an infinite cardinal, is there a chain ${cal C}subseteq {cal P}(kappa)$ such that $|{cal C}|=2^kappa$?