I just wanted to notify here that I have already reported this request for adding a Knife tool to Inkscape. Apparently someone else had already reported it but under the name of "Razor". People interested may contribute to my newer (and hopefully more detailed) request.

I'm sorry to say that I never investigated the formal request, until I had to decide whether to approve the first reply (by a new member) (which is only tentatively approved).

Anyway, I just wanted to note that you could do something like this with the Eraser tool already, set to narrowest width, which is probably 1 px. Or if you don't want a 1 px gap, you could just place a node where you want the path to break, and then use "Break path at selected nodes" button on Node tool control bar. Unless there is some other use for this kind of tool, that I haven't imagined, I'm not sure how much better it is than my suggestions. It seems mostly redundant. Maybe it would be faster and more convenient for some, who perhaps want to break a path "on the fly" and aren't concerned with precision? But it would be harder to draw a path which breaks another as any precise point, imo, than to place a new node and break there.

All of the above, my opinion