A continuum is a compact, connected, metrizable space.

What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of connected sets?

The following is an example of a contractible nowhere locally connected metrizable space, but I'm not aware of any compact examples (every segment on the "main line" has segments sticking out on a dense set).

This question was previously asked on MSE but didn't get an answer there.