First of all, the notion of $cone$ is a purely algebraic stuff, and not a metrical one. The $cone$ is naturally defined in the framework of $linear$ spaces, and not of Banach spaces. One can introduce, e.g., various "natural'' cones in a Hilbert space, without using its norm.

However, if $\left(X,\, d\right)$is a complete metric space, and $\psi:X\rightarrow\left[\,0,\,\infty\right)$is a lower semicontinuous function, then the partial ordering on $X$ defined by $x\preccurlyeq y$ iff $d\left(x,y\right)\leq\psi\left(y\right)-\psi\left(x\right)$is very useful in proving the Caristi-Kirk Fixed Point Theorem.

Another metrical variant would be to use the Ralph DeMarr' cone http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pjm/1103034358&view=body&content-type=pdf_1, combined with an Arens-Eells embedding.

First of all, the notion of $cone$ is a purely algebraic stuff, and not a metrical one. The $cone$ is naturally defined in the framework of $linear$ spaces, and not of Banach spaces. One can introduce, e.g., various "natural'' cones in a Hilbert space, without using its norm.

However, if $\left(X,\, d\right)$is a complete metric space, and $\psi:X\rightarrow\left[\,0,\,\infty\right)$is a lower semicontinuous function, then the partial ordering on $X$ defined by $x\preccurlyeq y$ iff $d\left(x,y\right)\leq\psi\left(y\right)-\psi\left(x\right)$is very useful in proving the Caristi-Kirk Fixed Point Theorem.

Another metrical variant would be to use the Ralph DeMarr' cone https://projecteuclid.org/euclid.pjm/1103034358, combined with an Arens-Eells embedding.