Who are some asexual characters in fiction (including ambiguously asexual characters, or characters that don't clearly identify as asexual)?

One particular "bachelor girl" (as her author termed her) stands out, but the Asexual Visibilty and Education Network mentions a number of characters who are often perceived as particularly sexless, including...
  • Bilbo Baggins in The Hobbit by J.R.R. Tolkien
  • Dr. Susan Calvin in I, Robot by Isaac Asimov
  • Lily White in Lily White by Susan Isaacs
  • Mr. Beebe and Cecil Vyse in A Room with a View by E.M. Forster
  • Mona Aamons Monzano in Cat's Cradle by Kurt Vonnegut
  • Sherlock Holmes (already mentioned by the OP)
  • Every character in Herland by Charlotte Perkins Gilman
(EDIT: I don't agree with all -- or even most -- of these, but, having read the books, I can understand why the characters might be seen as sexless.)

A sure standout asexual in classic literature, though, is Sue Bridehead, the protagonist's love interest in Thomas's Hardy's Jude the Obscure, "the ethereal, fine-nerved, sensitive girl, quite unfitted by temperament and instinct to fulfil the conditions of the matrimonial relation." (All quotes are from the novel.) Sue loves the idea of love, particularly of being loved, but not the intimacy that love entails.

As Jude puts it:

"People go on marrying because they can't resist natural forces, although many of them may know perfectly well that they are possibly buying a month's pleasure with a life's discomfort."

And as he goes on:

But you, Sue, are such a phantasmal, bodiless creature, one who-if you'll allow me to say it-has so little animal passion in you, that you can act upon reason in the matter, when we poor unfortunate wretches of grosser substance can't."

But Hardy doesn't necessarily condemn Sue's sexlessness, even though it tortures his poor protagonist; the author often compares her to an untouchable goddess, such as Venus Urania (the heavenly or spiritual aspect of the goddess of love).

Sue's surname itself seems a juxtaposition of "bride" and "maidenhead," and her ethereal asexuality is presented in stark contrast to the farm girl (and Jude's first wife) Arabella's ensnaring sensuality. She harbors romantic but non-physical tenderness towards Jude, even while knowingly tormenting him:

"I think I would much rather go on living always as lovers, as we are living now, and only meeting by day."

Sue eventually marries Phillotson, but sleeps apart from him. She ventures the following scenario to Jude:

"Wouldn't the woman, for example, be very bad-natured if she didn't like to live with her husband; merely"-her voice undulated, and he guessed things-"merely because she had a personal feeling against it-a physical objection-a fastidiousness, or whatever it may be called-although she might respect and be grateful to him? I am merely putting a case. Ought she to try to overcome her pruderies?"

Her aversion to what she calls "relations based on animal desire" come to a peak when her husband accidentally touches her -- and she jumps out the bedroom window.

When she leaves Phillotson, the lack of sex in the new relationship between Jude and Sue is perhaps more of a source of tension than even their marriages to other people.

"Sue, sometimes, when I am vexed with you, I think you are incapable of real love."

Eventually, Sue's (passionless? passionate?) sex with Jude leads to children, and tragedy.

And, in the end, Sue uses sex as the severest punishment she can inflict upon herself.
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