In Surah 18 of the Quran, the story of Zul-Qarnain - who is Alexander the Great - is recounted, and in it, he is said to have found the place where the sun rises and sets, and it is said that "it set in a spring of murky water":
Surah 18:83-91
83 They ask thee concerning Zul-qarnain. Say, "I will rehearse to you something of his story."
84 Verily We established his power on earth, and We gave him the ways and the means to all ends.
85 One (such) way he followed,
86 Until, when he reached the setting of the sun, he found it set in a spring of murky water: Near it he found a People: We said: "O Zul-qarnain! (thou hast authority,) either to punish them, or to treat them with kindness."
87 He said: "Whoever doth wrong, him shall we punish; then shall he be sent back to his Lord; and He will punish him with a punishment unheard-of (before).
88 "But whoever believes, and works righteousness,- he shall have a goodly reward, and easy will be his task as We order it by our Command."
89 Then followed he (another) way,
90 Until, when he came to the rising of the sun, he found it rising on a people for whom We had provided no covering protection against the sun.
91 (He left them) as they were: We completely understood what was before him.
The near-unanimous Muslim rebuttal to a critique of this passage is to assert that it means that from the perspective of Zul-Qarnain, the sun appeared to set in a spring of water.
There are three reasons that this is unlikely:
Regarding early Tafsir, both Ibn Abbas (619-687 AD) and al-Tabari (923 AD) seem to interpret the passage literally:
Tafsir Ibn Abbas - 18:86
(Till, when he reached the setting place of the sun) where the sun sets, (he found it setting in a muddy spring) a blackened, muddy and stinking spring; it is also said that this means: a hot spring, (and found a people thereabout) these people were disbelievers: (We said: O Dhu'l-Qarnayn!) We inspired him (Either punish) either kill them until they accept to believe that there is no deity except Allah (or show them kindness) or you pardon them and let them be.
Tafsir Al-Tabari - 18:86
The meaning of the Almighty's saying, "Until he reached the place of the setting of the sun he found it set in a spring of murky water," is as follows:
When the Almighty says, "Until he reached," He is addressing Zul-Qarnain. Concerning the verse, "the place of the setting of the sun he found it set in a spring of murky water," the people differed on how to pronounce that verse. Some of the people of Madina and Basra read it as "Hami'a spring," meaning that the sun sets in a spring that contains mud. While a group of the people of Medina and the majority of the people of Kufa read it as, "Hamiya spring" meaning that the sun sets in a spring of warm water. The people of commentary have differed on the meaning of this depending on the way they read the verse.
Later commentators, such as Ibn Kathir (1373 AD), and all modern commentators, take the "perspective" approach, as obviously, by that time there would have been no toleration for a belief that the sun "set" in any actual location. Earlier commentators, however, do not show these reservations, and go with the plain reading of the text.
Then, regarding the narration attributed to Muhammad:
Sunan Abi Dawud 4002
Narrated Abu Dharr:
I was sitting behind the Messenger of Allah (ﷺ) who was riding a donkey while the sun was setting. He asked: Do you know where this sets? I replied: Allah and his Apostle know best. He said: It sets in a spring of warm water.
The above unambiguously takes the literal reading of Surah 18, and if it is genuine, would be indisputable proof that Muhammad was mistaken about the setting of the sun, and nature of the world, and that he put that erroneous worldview into the Quran. However, there is some controversy over Al-Abani's (1914-1999 AD) rating of Sahih-in-chain for the Hadith, with some saying it should be weak.
This should not be surprising, as obviously, to a Muslim, this Hadith simply cannot be genuine. It disproves Islam. Regardless, it is a testimony to an early tradition within the Islamic community of a literal understanding of the passage in question from Surah 18.
There are also narrations in which Muhammad seemed to teach that the sun orbited the earth once per day, and that during the day, it was daytime for the whole earth simultaneously, indicating a flat-earth view.
For instance:
Sahih al-Bukhari 3199
Narrated Abu Dhar:
The Prophet (ﷺ) asked me at sunset, "Do you know where the sun goes (at the time of sunset)?" I replied, "Allah and His Apostle know better." He said, "It goes (i.e. travels) till it prostrates Itself underneath the Throne and takes the permission to rise again, and it is permitted, and then (a time will come when) it will be about to prostrate itself but its prostration will not be accepted, and it will ask permission to go on its course but it will not be permitted, but it will be ordered to return whence it has come, and so it will rise in the west. And that is the interpretation of the Statement of Allah: "And the sun Runs its fixed course For a term (decreed). That is The Decree of (Allah) The Exalted in Might, The All-Knowing." (Surah 36:38)
This is also narrated in Sahih al-Bukhari 4802-03, 7433, and Sahih Muslim 159a-d, and it is reiterated that this is Muhammad's commentary on Surah 36:38.
The above appears to teach a flat-earth view of the world, wherein the sun "goes" somewhere at night, for all of the earth. In reality, the sun does not "go" anywhere at sunset, because it is always sunset somewhere on earth. So, it would not make sense to say the sun "goes" to prostrate under the throne of Allah during the nighttime, as it is always nighttime somewhere on earth, and so the sun would be in constant "prostration", making Muhammad's statement here rather meaningless.
Similarly, Muhammad taught that when the sun is directly overhead during the day, "the gates of Hell are opened and it is stoked up", and therefore, prayer during that time is prohibited:
Sunan an-Nasai 572
Abu Yahya Sulaim bin 'Amir, Damrah bin Habib and Abu Talhah Nu'aim bin Ziyad said:
"We heard Abu Umamah Al-Bahili say: 'I heard 'Amrah bin 'Abasah say: I said: 'O Messenger of Allah, is there any moment which brings one close to Allah than another, or any moment that should be sought out for remembering Allah? He said: 'Yes, the closest that the Lord is to His slave is in the last part of the night, so if you can be among those who remember Allah at that time, then do so. For prayer is attended and witnessed (by the angels) until the sun rises, then it rises between the two horns of the Shaitan, that is the time when the disbelievers pray, so do not pray until the sun had risen to the height of a spear and its rays have disappeared.
Then prayer is attended and witness (by the angels) until the sun is directly overhead at midday, and that is the time when the gates of Hell are opened and it is stoked up. So do not pray until the shadows appear. Then prayer is attended and witnessed (by angels) until the sun sets, and it sets between the horns of a Shaitan, and that is the time when the disbelievers pray.'"
However, the sun is always directly overhead of someone, somewhere on earth, because the earth is round. Therefore, the "gates of Hell" would be opened at all times, if that is when they open.
Finally, this prohibition on prayer based on the location of the sun is also seen in the following narration:
Sahih Muslim 828b
Ibn 'Umar reported Allah's Messenger (ﷺ) as saying:
Do not intend to observe prayer at the time of the rising of the sun nor at its setting, for it rises between the horns of Satan.
The fact that the sun rises between the horns of Satan is further narrated Sunan Ibn Majah 1253, Sunan Abi Dawud 1277, and Sunan an-Nasai 572.
Once again, if the sun "rises between the horns of Satan", and therefore no one can pray when it rises or sets, then no one can pray at all, because the sun is always setting and rising, somewhere on earth.
Taken together, these narrations indicate that Muhammad believed that sunrise and sunset were events that happened for all of humanity at roughly the same time, as the sun travels across a flat earth on a circuit each day. These narrations all make sense in such a flat-earth model. However, they do not make sense on a round-earth model.
Also, it should be noted that God, in the Bible, never prohibited prayer at any time of day, at any point, for any reason. Instead, we are told to "pray without ceasing" (1 Thessalonians 5:17, also Psalm 55:17, etc.).
Muhammad said many things which do not make sense given the fact that the earth is round, and orbits the sun. Instead, he appeared to hold to a flat-earth view of the world, wherein the sun rose and set once per day for all of the earth. He also seemed to believe that the sun descended through a pool of murky water somewhere in the West of the planet at sunset, and that a Muslim man named Zul-Qarnain traveled to the location where it happens.