[I'm sure there are a few other very good threads on this subject, but I'll let someone else post those]

The short answer is yes, there was a Palestinian Identity prior to the founding of the state of Israel. The longer answer takes on a more interesting question to my mind: how was that identity different than the Palestinian Identity that existed after 1949 (or 1956, 1967, 1972, 1987 or 1991 for that matter)?

So for starters, if we take as identity in this case the understanding that people would self-identify as belonging or of a certain place, however fuzzy the geography, then we can say with near-absolute certainty two things about Palestinian Identity in the Ottoman period (ie: from at least 1516 to WWI, so before and during the period of the Yishuv), 1) There was such a thing as Palestine and 2) people identified as being from Palestine. Now, many people are quick to point out that there was no Ottoman province of Palestine (the same people are also quick to point out that there was no such thing as "Iraq" prior to the end of WWI, but that's another argument), and that is a very narrow definition of whether or not a place "existed". For sure the Ottomans divided what is now Israel/Palestine into a number of different administrative districts based around regional cities (Damascus, Haifa, Jerusalem, and so on), but one only needs to move outside of the archives of government administration and into the realm of the press, literature and philosophy to understand that people used the term "Filistin" or "Filistini" very frequently to both identify themselves, and to refer to a very particular place. And moreover, this place or identification was not limited to Muslims. Christians and Jews would often share this identity, and especially so in the wake of the 1908 Ottoman Constitutional Revolution, in which people of the three religions (including some future leaders of the Zionist movement) came together to represent the various districts of Palestine in the Ottoman Parliament. Much as people of these three faiths shared similar cultural heritages (literature especially) in multi-confessional cities like Baghdad or Alexandria, the people of Palestine had much in common in terms of a shared cultural memory, history, literature, food, music, etc.

What sort of evidence do we have? I have mentioned literature, culture and music as one set of sources, but there are also maps that attempt to define Palestine, or at least depict it, from the Ottoman period. We also can point to a number of manuscripts that clearly mention "Palestine" as a place, and it can be assumed that the people in such a place may be considered "palestinian".

That said, the important caveat about identity in the Ottoman period is that it certainly was a multi-layered thing. People who might identify as Palestinian might also identify as "Ottoman" or "Damascene" or "Christian" or "Peasant" or "Seaman" depending on who is asking and when. And this goes for pretty much any area of the Ottoman empire. It's only after the rise of nationalism that people start using the state (imagined or real) as a primary marker of identity.

Things only begin to change rapidly in terms of Palestinian Identity after the establishment of the British Mandate following WWI. Unlike the prior Ottoman administration (which had its problems, but at least put the various groups on roughly equal footing), the British system set up an open antagonism between zionist settlers and local Muslim and Arab leaders. As such, the identity of "Palestinian" became increasingly defined in opposition to Zionism. Whereas the Palestinian Identity of the Ottoman period was mutli-confessional and tied primarily to the land and shared culture, it now became re-defined as Arab, primarily Muslim and in opposition to zionist and British incursion. This identity marker is solidified following the establishment of the state of Israel, but the exact parameters have been affected in numerous ways following the various armed conflicts of the Cold War period and the ongoing settlement activities (and violent reactions to such activities) in the 21st century.

Some further reading on this subject:

Rashid Khalidi Palestinian Identity

Michelle U. Campos Ottoman Brothers

Salim Tamari, Year of the Locust

Wasif Jawharriyeh The Storyteller of Jerusalem

Arieh Bruce Saposnik Becoming Hebrew

I think it's important, for the sake of this thread, to identify whether you're refering to an identity of a Palestinian state, or a unified and accepted understanding of a people as being "Palestinian"