= Solution
Take the <connected space>[topologist's sine curve]
$$
S=\{(x,\sin(1/x)):0<x\leq1\}
\cup(\{0\}\times[-1,1]).
$$
It is closed and bounded, hence compact. The oscillating graph is connected and accumulates on every point of the vertical segment, so its closure $S$ is connected. No path joins the graph to the vertical segment, as continuity of the first coordinate and the unbounded oscillation of $\sin(1/x)$ would fail at the first time the path reaches $x=0$. Thus $S$ is not path connected.
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