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Yes. A solenoid is a homogeneous continuum (=compact connected metric space) embeddable in $\mathbb{R}^3$ that is not locally connected at any point, in fact, a small neighborhood of each point looks like the Cantor set crossed with an interval. Its generic projection to $\mathbb{R}^2$ is compact, connected, and not locally connected at each of its points. [Edit I am not longer confident that the last claim is true.]