I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire.
Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category theorem: local compactness or complete metrizability or Čech completeness because all these properties are hereditary with respect to closed subspaces.
An interesting example of a Baire metrizable space that fails to be completely metrizable is given in an answer to What are some motivating examples of exotic metrizable spaces. This example contains $\mathbb Q$ as a closed subspace.
It would be nice to see some further examples.
Added: I would prefer to have examples of high regularity (at least Hausdorff, preferably Tychonoff).