Can the word Torah include other books in the Tanakh? If so, are there examples of this in Jewish literature (ancient rabbinical literature, medieval, etc)
There are many examples of the Talmud referring to Tanach as the Torah. One example is Sanhedrin 37a:
והיינו דאמר ליה ההוא מינא לרב כהנא אמריתו נדה שרי לייחודי בהדי גברא אפשר אש בנעורת ואינה מהבהבת אמר ליה התורה העידה עלינו סוגה בשושנים שאפילו כסוגה בשושנים לא יפרצו בהן פרצות
And this is like an incident involving Rav Kahana, as a certain heretic said to Rav Kahana: You say that it is permitted for a menstruating woman to seclude herself with a man, i.e., her husband. Is it possible to set fire to chips of kindling and not have them blaze and burn? How can the couple be relied upon not to engage in sexual intercourse? Rav Kahana said to him: The Torah testifies concerning us that we are “set about with lilies” as the Jewish people do not breach even a fence made of lilies.
The verse brought is from Song of songs.
This phenomenon is noticed by the Rashbatz in his Zohar HaRakiah Mitzvos Aseh § 19 s.v. וההלל תגמור. Although, he is discussing Rabbinic enactments being referred to as Torah. He says the gemarra sometimes uses the word תורה to refer to something not in the Torah, if it’s הוראה וענין קבוע, although I'm not sure what he means by that. Perhaps it could apply to Tanach as well.
Rashi Genesis 44:8. Rashi (citing Bereishis Rabba 92:7) says there are ten uses of kal vachomer logic arguments in the "Torah."
Behold, the money, etc.: This is one of the ten a fortiori conclusions (deductions from minor to major) mentioned in the Torah, and they are all enumerated in Gen. Rabbah (92:7). 10. הן כסף אשר מצאנו: זה אחד מעשרה קל וחומר האמורים בתורה. וכולן מנויין בבראשית רבה (צב ז):
He means Tanach, as there are only three in the Torah. ("We brought back the silver given by mistake, you think we'd steal?!" "If her father shamed her she'd have to go into isolation, you think it's a lesser consequence with God?!" and "you rebelled against God even with a leader like Moses; good luck staying on the straight-and-narrow after his demise!")
Talmud Yoma 28b
אָמַר (רַב), וְאִיתֵּימָא רַב אָשֵׁי: קִיֵּים אַבְרָהָם אָבִינוּ אֲפִילּוּ עֵירוּבֵי תַבְשִׁילִין, שֶׁנֶּאֱמַר: ״תּוֹרוֹתָי״, אַחַת תּוֹרָה שֶׁבִּכְתָב וְאַחַת תּוֹרָה שֶׁבְּעַל פֶּה.
Rav said, and some say Rav Ashi said: Abraham our Patriarch fulfilled the entire Torah, even the mitzva of the joining of cooked foods, a rabbinic ordinance instituted later, as it is stated: My Torahs. Since the term is in the plural, it indicates that Abraham kept two Torahs; one, the Written Torah, and one, the Oral Torah. In the course of fulfilling the Oral Torah, he fulfilled all the details and parameters included
Further, Maimonides opens his introduction to his codification of Torah law, Mishneh Torah, with the statement (based on Talmud, Berachoth 5a): “All the mitzvoth that that were given to Moses at Sinai were given with their explanation. As it is written,4 ‘I will give the tablets of stone, and the Torah, and the mitzvah.’ ‘The Torah’—this is the Written Torah. ‘And the mitzvah’—this its explanation. And He commanded us to fulfill the Torah in accordance with the mitzvah. This ‘mitzvah’ is what is called the Oral Torah.” Indeed, the two are inseparable, as no text can have any authoritative meaning without an accompanying tradition as to what it means and what are the principles that govern its interpretation.
For a comprehensive explanation see. https://www.aish.com/atr/Torah_versus_Talmud.html?mobile=yes