From David Cullen@21:1/5 to All on Sun Apr 10 05:56:51 2016
Ok if you are asking if there is an example of a metric space which is connected, and at the same time is also totally path disconnected, then
I suspect the knaster-kuratowski fan space will suffice. It is known
to be connected, and I suspect it is totally path disconnected as well,
though I haven't done the legwork yet. Does this work?