Was the daughter of Pharaoh's arm lengthened miraculously?

I agree that not all midrashim are intended literally, but some are. To find out for certain, you need to check the sources. Also, some of what people call "midrashim", by which they mean "fanciful back story" or "legends or stories containing lessons or morals" are really nothing of the kind. One example is the famous story of the daughter of Pharaoh and how her arm lengthened miraculously when she reached for the basket containing the baby Moshe. According to one school of thought this interpretation is not a description of a historical event, but a legend, composed at some later date for the purpose of teaching us something.

Though I don't make any claims about what actually happened at the riverside between Moshe and the princess, and though I agree that the stretchy-arm interpretation was created long after the event it describes,  I'm absolutely certain that the author of this interpretation meant it literally. He wasn't trying to teach us something. He was trying to tell us what he thought had happened.

Read the rest of the post to find out my reasons for saying this.

The argument about princess's hand (to my knowledge) is first found on BT Sota 12A where R. Judah and R. Nechamia argue about the meaning of the word ama. According to R. Judah the word means yad, which translates as both hand and arm. According to R. Nechamia the word means "maidservant." To see why this matters just replace the word ama in Exodus 2:5 with either meaning:

Then Pharaoh's daughter went down to the Nile to bathe, and her attendants were walking along the river bank. She saw the basket among the reeds and sent her [ama] to get it.

There is no question at all that the original argument between R. Judah and R. Nechamia is about pshat, ie. the plain meaning of the verse. Did she send her arm or send a servant? However, as you've surely noted, this argument does not say anything about the alleged plasticity of the princesses's arm. Even per R. Judah, she may have stretched out her arm in an ordinary, non-miraculous manner.

However, there is more to the story. As the gemarah (by which I mean an anonymous Amora or Sevora) explains, R. Judah feels compelled to say that ama means hand/arm because of an oddity in the verse. For no apparent reason, we are told that the attendants were "walking along the river bank". Because this phrase seems unnecessary, it was deduced that what the verse really means to say is that all of the attendants were "walking toward death", that is, they were killed As R. Johanan says: "The word for 'walk' means nothing else than death; and thus it says (Genesis 25:32): "Behold Anochi holech lamut [=I am walking to die.] Explains the Gemarah:

When [the maidens] saw that she wished to rescue Moses, they said to her, 'Mistress, it is the custom of the world that when a human king makes a decree, though everybody else does not obey it, at least his children and the members of his household obey it; but thou dost transgress thy father's decree!' Gabriel came and beat them to the ground.[DB: to ensure that Moshe would be saved.]

So how could the princess have sent a servant to fetch the basket if all of them had been killed?  And furthermore, the first time the attendants are mentioned they are called naarot [= "attendants" above]. Why refer to one of them as an ama, and not as a naara later in the same verse?  To the close reading midrashist strange and cryptic word choices such as these all mean something.

But to another type of midrashist, it might not mean anything at all. Thus, we have the gemerah (by which I still mean an anonymous Amora or Sevora, though perhaps not the same one from above) seeking to defend R. Nechamia and the possibility that ama means maidservant. He argues:

...according to him who said that it means 'her hand', the text should have been yadah!

In other words, why doesn't the Torah say what the Torah means? If we're to understand that the princess fetched the basket with her own, non-stretching, ordinary arm, why didn't the Torah use the ordinary word for arm? Why doesn't it say yad if yad is what Torah means? Why use the unusual and ambiguous word ama instead?

Answers the Gemarah:

It teaches us that [her arm] became lengthened

That is, the word ama is used to convey that it was specifically her own ammah (that is the part of her body from her hand to her elbow) that became lengthened. Continues the Gemarah (paraphrase) The same thing happened when Og's teeth lengthened, and when Achasverosh's scepter lengthened. Though describing those two events is beyond the scope of this post, it seems clear from his language that the anonymous Amorah or Sevorah who interpreted the word ama in our verse as a very strong hint that the princesses's whole arm grew also thought that the same kind of miraculous lengthening had occurred on two other occasions.  He's not conveying to us secrets or deep messages or allegories. He's merely reading and translating the verse in the light of his understanding of the words and context and attempting to tell us what happened. That makes his interpretation a pshat reading.

In fact the whole issue might be even simpler. It might not hinge on accepting that our anonymous sage  believed that all three stretching events were literal, as the plain sense of his argument suggests. Rather it may come down to a straight reading of three words וַתִּשְׁלַ֥ח אֶת־ אֲמָתָ֖הּ. Our anonymous interpreter might have read the word amata as her hand because, as described above, he believed all of the royal attendants had been killed by Gabriel. Thus, the princess had no choice but to use her own hand. However, as he would have also realized, the verse's verb vatishlach [= and she sendt is not one typically used to modify a body part. You might send your servant, but would you send your own arm?

A solution is suggested by Ezekiel 44:20:

Neither shall they shave their heads nor suffer their locks to grow long they shall only poll their heads

The word used here for "grow" is yishalaychu, a form of the same verb that appears in our verse in Exodus, and as in Exodus, the verb here is used to modify a body part (hair) If that verb means "grow" in Ezekiel, might it also mean "grow" in Exodus? As a result וַתִּשְׁלַ֥ח אֶת־ אֲמָתָ֖הּ is read (not interpreted but READ) as "...and she grew her arm."

Understood in this way, saying that her arm grew is really no different than saying that the hail plague consisted of fire and ice intermingled. Both are miraculous events, unlike anything we have ever seen, and both (per the understanding of ama described here) seem to be the plain meaning of the verse. From this perspective, saying that her arm literally grew is no more difficult then saying that fire and ice literally mixed. Both are straight readings of the verses in question.

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