Can the infaillible prophets make mistakes?

In answer to the video "How Islam Shaped Its Sources"

As is explicit in the Quran, the divine protection of the carriers of the revelation pertains strictly to the revelation itself. But in everyday affairs, the messengers, who are still humans endowed with freewill and thus the potential, if not to sin due to their heightened level of spiritual awareness, to make mistakes, they are left to their own devices in their everyday lives to fight off the assaults of evil forces. No prophet was in a constant state of communication with the divine realm. The hadith and Quran itself speak of long periods where revelation had stopped, and the subsequent tauntings of his enemies on the issue, the questions of his followers and his anxious anticipation. 

The Quran never came to correct the prophet's worldviews in terms of knowledge of nature and general causality, neither of his contemporaries but rather guide him and the rest of humanity through him, to the most complete, advanced human spiritual knowledge. The divine protection  therefore only pertained to the Quran which is the source of that perfect spiritual knowledge. 

The prophet was "uswa hasana" in his application of the Quran, just as following Jesus' way, as he is quoted saying in the NT, meant following his footsteps in his application of the Torah. "The way" of Jesus Jn14:6 is outlined in Lk10:25-28 where he commands strict observance of Jewish laws. In that passage from Luke he is asked about the conditions of salvation and the questionner quotes from 2 passages. The first is Lev19 which details certains laws like the observence of the sabbath and admonishes to

"Keep all my decrees and all my laws and follow them. I am the LORD".

The 2nd passage quoted by the questionner is the second is Deut6 which speaks of loving the One God and obeying His commandements

"keep the commands of the LORD your God and the stipulations and decrees he has given you. Do what is right and good in the LORD's sight..obey all this law before the LORD our God, as he has commanded us, that will be our righteousness".

As one can clearly see, one is justified before God, not by faith alone but by deeds too. Consequently the Nazarenes, Jesus' early group of small band of followers, observed all Jewish customs outlined in the Torah but differed from Jews in that they recognized Jesus as the Messiah. The Nazarenes grew among the Israelites but persecutions forced them to go into hiding, with Paul playing a central role in their persecution prior to his convertion. After he joined their ranks, he started influencing the group leaders, namely Peter and James, to reach out to Gentiles. With more non-Jews entering the fold, many Jewish customs were abandonned Acts15:1-29 and the Nazarenes who were centered in Jerusalem gradually became isolated. The main Christian movement started looking up to Paul for leadership, instead of Jesus' brother James, a strict observer of Jewish Law and considered as Jesus' successor in non-canonical Gospels.

With the establishment of Christianity as a state religion in Rome by Constantine in the 4th century, this small original band definetly fled Jerusalem, in the surrounding deserts and managed to survive outside Palestine as they are mentionned by Jerome upto 380AD to have lived in the Syrian desert. Among them the Ebionites (who claimed to descend from the original Jewish disciples led by James) and Elchasites who rejected Paul as a charlatan and his teachings as falsehood, as well as the Zadokites, Essenes, Rechabites, Sabeans, Mandaeans etc. They had their own writing which they considered scripture, composed of an oral tradition attributed to Jesus, and some HB books. Their writings are known, among others as Gospel of the Nazareans, Gospel of the Hebrews and Gospel of the Ebionites. They would later write that Paul was a false apostle who taught heresy based on the fact he was a failed convert who was disappointed with Judaism and therefore motivated to teach against its laws (circumcision, kashrut, etc..). 

Unfortunately the group that opposed them and their practices gained more converts, obviously as it appealed much more to non-Jews, more particularily the hellenized Romans and Greeks. The Nazarenes and similar groups were inevitably marginalised while the more and more dominant groups decided what the Church’s organizational structure would be, as well as its official creeds, or which books would be accepted as Scripture. The group that became "orthodox", further sealed its victory, by the pens of early writers like Iraeneus Justin Martyr and Tertullian, claiming that it had always been the majority opinion of Christianity, Jesus and his apostles.