Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected subset.

The answer is negative if the space is assumed to be connected and locally path-connected. Since every component of a connected and locally path-connected space is path connected.

Added after some useful comments: If we assume that the space X is actually a metric space (together with the metric topology), then can it possible to contain non-trivial path-connected subset. Note that i assume that any component of the metric space X is non-trivial (not a point).