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My question is the following:

Does there exist a connected metric space $\ X,\ $ where $\ |X|>1,\ $ which contains no separable connected subspace $\ Y\ $ with $\ |Y|>1\ $?

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The answer is yes, and an example can be found in

Simon, Petr: A connected, not separably connected metric space, Rend. Istit. Mat. Univ. Trieste 32 (2001), suppl. 2, 127–133.

Quoting from the introduction:

A separably connected space is a topological space, where every two points may be joined by a separable connected subspace. We present an example of a connected, but not separably connected metric space and of a connected metric space, which contains no connected separable subspaces other than one-point ones.

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