It remains in use colloquially; for example, as a counterpart to "gentleman," in the phrase "ladies and gentlemen," and is generally interchangeable (in a strictly informal sense) with "woman." (e.g., "The lady at the store said I could return this item in thirty days."). "Ladies" is also the normal text on the signs to any female toilet in a public place in the UK, again paired with "Gentlemen" (or "Gents").
White's anecdote touches on a phenomenon that others have remarked on as well. In the late nineteenth and early twentieth century, in a difference reflected in Nancy Mitford's U vs. non-U distinction, lower class women strongly preferred to be called "ladies" while women from higher backgrounds were content to be identified as "women." Alfred Ayers remarked in 1881 that upper middle class female store clerks in his day were content to be "saleswomen," while lower class female store clerks, for whom their job represented a social advancement, indignantly insisted on being called "salesladies." Something of this sense may also be underneath Kipling's lines:
These social class issues, while no longer on the front burner in the twenty-first century, have imbued the formal use of "lady" with something of an odour of irony (e.g: "my cleaning lady").
