There are several 7th c non-Muslim accounts that are thought to describe contemporary assessments of Muhammad during his lifetime. One of these is the "Doctrina Jacobi nuper baptizati."
Let's look at a well-known theory by Crone and Cook, and a counter-theory by Yehûdā Nevô.
Doctrina Jacobi (D. Jacobi) is set in the year 634 CE, and probably written within a few years of this date. The relevant passage purports to be a letter from a Palestinian Jew named Abraham: https://en.wikipedia.org/wiki/Teaching_of_Jacob
Crone and Cook (1977) accept this as a reference to Muḥammad - or rather, that it preserves an older, more historical set of facts about Muḥammad, which later Islam suppressed because of its Judeo-Messianism.
Specifically, they point out this Saracen prophet was alive at the time of the invasion, personally led it, proclaimed himself to be the herald of the (Jewish) Messiah, and claimed to hold the keys of Paradise.
Nevô (2003) asks, why should one identify the reference in D. Jacobi with Muḥammad? If this prophet in question existed, he fit the general Judaic and Christian monotheistic background.
Moreover, it is more likely that this message was proclaimed in Aramaic than in Arabic, for if his proclamations worried people they must have been in a language widely understood by both the Jews and Christians in the area.
Any widely known theological message behind a conquering prophet would likely be in the language of the conquered. If so, the group he was associated with had arisen in an Aramaic speaking community - which would mean a local (not Hijazi) one.
Thirdly, D. Jacobi tells us little about the group's religion. It could have been any form of monotheism; and indeed it accords much more with Jewish, Judeo-Christian, or even Christian belief than it does with Islam.
On all these counts, argues Nevô, D. Jacobi provides no support for the identification of this prophet with Muḥammad. In fact if one thing is clear, it is that the account in the D. Jacobi does not describe the Muḥammad we know from any other tradition.
Crone and Cook acknowledge that "this testimony is of course irreconcilable with the Islamic account of the Prophet's career" - and conclude from this fact that we have here older material - "a stratum of belief older than the Islamic tradition itself" ..
.. which "proves" that the true, historical Muḥammad led the invading Arabs, proclaimed the advent of the Messiah, and claimed to hold the keys of Paradise.
However, the only reason for regarding this material as the original, true version of Islamic history is the a priori identification of the prophet here mentioned as Muḥammad.
Yet if this material proves anything, says Nevô, it is that the prophet of the D. Jacobi can easily be almost anybody other than Muḥammad. Prophethood was, to use Wansbrough's (1987) term, a monotheistic constant.
In those troubled years of Late Antiquity and into the 7th c, contemporary accounts reveal that there was no shortage of such prophets, appealing to the various Christian and Jewish sects.
It should be noted that Nevô provides this counter-argument within a larger thesis about the transition from Byzantine to Arab rule in the 7th c. His theory was that the Arab conquest was a fiction, or rather a legendary redaction.
This redaction was composed in the 2nd half of the 7th c by Arabs who merged 3rd hand memories of the Byzantine-Persian wars with 2nd hand memories of their peaceful acquisition of Byzantine territory.
So, any early account of a conquering prophet is particularly suspect by Nevô. All texts that fit this descriptive category, including Sebeus (661), Secrets of Rabbi Simon ben Yoḥay (mid 8th c), and D Jacobi (634) above, are evaluated and dismissed as redactions or mis-ID's.
Furthermore, Brock (1982) reminds us that only a few late chronicles provide any details of Muḥammad's early career - those after the Traditional Account had already been formulated by Arab authorities.
Christians who came into contact with the newcomers and lived side buy side with them did not, apparently, learn from them anything of Islam for more than two generations. Byzantine and Syrian literature displays no knowledge of Islamic teachings until the early 8th c.
Syriac authors recognize Islam as a new religion only late in the day; as Brock points out, "it was perhaps only with Dionysios of Tellmaḥre (d. 231/845) that we really get a full awareness of Islam as a new religion."
In fact, not one early Syriac or Greek source describes the Arabs of the early 7th c as Muslims, or in terms we can recognize as conclusively Islamic. Various reasons for this have been proposed, Crone and Cook providing the Jewish Messianic-to-Arabic Prophet theory.
However, Nevô concludes that no theory fits the case as well as the simple proposition that the 7th c Jews and Christians did not discern Islam because it was not there to discern.
Crone, Patricia, and Michael Cook. Hagarism: the Making of the Islamic World. Cambridge Univ. Press, 1977.
Nevô Yehûdā D., and Judith Koren. Crossroads to Islam: the Origins of the Arab Religion and the Arab State. Prometheus Books, 2003.
Brock, Sebastian P. (1982). "Syriac Views of Emergent Islam." In Studies on the First Century of Islamic Society, edited by J. Y. N. Boll. Carbondale, Ill.: Southern Illinois University Press, 1982.