As we already know, the metric of discrete metric space cannot be induced by norm. So the derivative cannot be defined even with respect to Frechet's or Gateux' definition. From analysis on metric space we can define a kind of metric derivative by (From Ambrosio's Topics on Analysis in Metric Spaces)
Definition. Given a curve $\gamma:[a,b]\to E,$ we define the metric derivative of $\gamma$ at the point $t\in(a,b)$ as the limit $$\lim\limits_{h\to 0}\frac{d(\gamma(t+h),\gamma(t))}{|h|}$$ whenever it exists and in this case, we denote it by $|\dot{\gamma}|(t).$
So my question is, can such definition still hold for discrete metric space?
Of course you can still make this definition. All the definition requires is that $E$ is some metric space, so that $d(\gamma(t+h),\gamma(t))$ is meaningful.
However, if $E$ is discrete, then it is a rather useless notion, since every curve in $E$ is constant and so has derivative $0$.