Note: As the author of the question - I feel that a stronger answer would be to point to historical evidences outside of Scripture, and evidences earlier than the Talmud. For example, evidences regarding Beit Hillel would be helpful.
1. Question Restatement :
What are the earliest textual evidences that a day in ancient Israel began at sunrise?
Answers : Section #2. Rabbinic Literature; Section #3. the New Testament ...
This answer presupposes a Christian authority to challenge and disregard Rabbinic authority and tradition :
NASB, Titus 1:13-14 - This testimony is true. For this reason reprove them severely so that they may be sound in the faith, 14 not paying attention to Jewish myths and commandments of men who turn away from the truth.
Note: These texts are generally accepted as reasonable bases to question whether a day was reckoned from sunset, or sunrise - objections notwithstanding.
2. Rabbinic Literature :
MISHNA Pesahim IV - And the Sages say, "In Yehuda, they would do work on the eve of Pesach until noon; and in the Galilee they did not work at all [on that day]." And [with respect to] the evening [of the fourteenth of Nissan in places like the Galilee], Beit Shammai forbids [work], but Beit Hillel permits [it] until the sunrise.
B. Talmud, Book 9 - Tracts Maccoth, Shebuoth, Eduyoth, Abuda Zara, and Horioth - R. Simlayi lectured: Six hundred and thirteen commands were said to Moses; three hundred and sixty-five of them negatives, corresponding to the number of days in a year counting according to sunrise; and two hundred and forty-eight positives,
B. Talmud, Chulin 83a - GEMARA. Our Rabbis taught: This was expounded by R. Simeon b. Zoma: Since the whole passage deals only with the laws concerning consecrated animals,  and with regard to consecrated matters [a day means] the day and the night following it ... [28 For in connection with the eating of sacrificial meat it is written (ibid. VII, 15). It shall be eaten on the day of his offering; he shall not leave any of it until the morning. Thus it may be eaten the whole of the night following the day.]
3. The New Testament :
NASB, Matthew 27:57-62 - 57 When it was evening, there came a rich man from Arimathea, named Joseph ... 59 And Joseph took the body [Mark 15:46 bought a linen cloth] ... and wrapped it in a clean linen cloth ... 62 Now on the next day, the day after the preparation,
NASB, Matthew 28:1 - Now after the Sabbath, as it began to dawn toward the first day of the week, Mary Magdalene and the other Mary came to look at the grave.
Greek Interlinear, Luke 23:54 - Indeed, it was the Day of Preparation, and Sabbath dawning.
NASB, John 20:19 - So when it was evening on that day, the first day of the week ...
4. The Old Testament
Genesis 1:6-8 - Then God [made the sky] and there was evening (ending the daytime), and there was morning (ending the nighttime), [making] the second day.
Note: From Rabbi Samuel ben Meir's Peshat of Genesis 1:8, (@emory.edu) and Can The Torah Contradict Halacha (Jewish Law)?)
NASB, Genesis 19:34 - On the following day, the firstborn said to the younger, “Behold, I lay last night with my father; let us make him drink wine tonight also; then you go in and lie with him,
NASB, Exodus 16 - 23 then he said to them, “This is what the Lord meant: Tomorrow is a sabbath observance, a holy sabbath to the Lord. Bake what you will bake and boil what you will boil, and all that is left over put aside to be kept until morning.” 24 So they put it aside until morning, as Moses had ordered ... 27 It came about on the seventh day that some of the people went out to gather, but they found none.
NASB, Leviticus 7:15 - ‘Now as for the flesh of the sacrifice of his thanksgiving peace offerings, it shall be eaten on the day of his offering; he shall not leave any of it over until morning.
NASB, Leviticus 22:30 - It shall be eaten on the same day, you shall leave none of it until morning; I am the Lord.
NASB, Leviticus 23:26-32 - 26 The Lord spoke to Moses, saying, 27 “On
exactly the tenth day of this seventh month is the day of atonement; 32 It is to be a sabbath of complete rest ... on the ninth of the month at evening, from evening until evening you shall keep your sabbath.”
Note: If presupposing "calendar days according to evenings" - then this passage would exclude the daytime of the 10th - internally contradicting itself.
NASB, Numbers 33:3 - They journeyed from Rameses in the first month, on the fifteenth day of the first month; on the next day after the Passover the sons of Israel started out boldly in the sight of all the Egyptians,
NASB, Deuteronomy 21:22 - “If a man has committed a sin worthy of death and he is put to death, and you hang him on a tree, 23 his corpse shall not hang all night on the tree, but you shall surely bury him on the same day (for he who is hanged is [s]accursed of God), so that you do not defile your land which the Lord your God gives you as an inheritance.
NASB, 1 Samuel 19:10-11 - 10 ... And David fled and escaped that night. 11 Then Saul sent messengers to David’s house to watch him, in order to put him to death in the morning. But Michal, David’s wife, told him, saying, “If you do not save your life tonight, tomorrow you will be put to death.”
NASB. 1 Samuel 20 - 18 Then Jonathan said to him, “Tomorrow is the new moon [first of the month], and you will be missed because your seat will be empty. 19 When you have stayed for three days, you shall go ... 34 Then Jonathan ... did not eat food on the second day of the new moon [the month] ... 35 Now it came about in the morning that Jonathan went out into the field for the appointment with David [on the third of the month].
NASB, 2 Chronicles 35 - 1 Then Josiah celebrated the Passover ... and they slaughtered the Passover animals on the fourteenth day of the first month. ... 10 So the service was prepared ... 11 They slaughtered the Passover animals ... 13 So they roasted the Passover animals ... 16 So all the service of the Lord was prepared on that day to celebrate the Passover, and to offer burnt offerings on the altar of the Lord according to the command ...
NASB, Jonah 4:7 - But God appointed a worm when dawn came the next day and it attacked the plant and it withered.
... etc, etc.
Note: See also a compelling argument at http://www.iahushua.com/ST-RP/twt.htm; But, the primary reference needs confirmation, (RABBINICAL ESSAYS BY JACOB Z. LAUTERBACH HEBREW UNION COLLEGE PRESS CINCINNATI 1951).