The correct answer is "further." In English, "further" is used when requesting more detailed information or additional data. It often refers to a greater degree or extent of something, especially concerning abstract concepts like information or discussion, rather than physical distance. Here's why the other options are less appropriate:
farther
Typically used for physical distance.
faster
Refers to speed, not to the level of detail or amount of information.
furthur
Is a misspelling of "further."
Thus, the sentence should read: "Can you give me further details, please?"