The short answer is overwhelmingly yes. However, the formation of the identity, and how strong it was before the 1948 Israeli Declaration of Independence, is still a hotly contested issue today.

To give you some background on why that is, the question of "Palestinism", or Palestinian nationalism, is extraordinarily relevant to Israel's negotiations over the "right of return" for refugees of the 1948 war. The question of how distinct the area felt itself to be also creates an idea that Israel fears: namely a justification for a Palestinian state where Israel is today. This fear was much more pronounced following 1948, and Israel (and Jordan) undertook efforts to educate Palestinians within their borders on how Palestinism was a recent invention. I'll expand on this later.

In order to understand the Palestinian independence movement, I'll break this up into 3 parts: pre-WWI Palestinism, post-WWI Palestinism, and post-1948 Palestinism.

The phrase Palestinism is one I've read only in a few works, and is really just an ad-hoc word to simplify things. If you haven't heard the word I completely understand, just be aware I'm referring to "Palestinian nationalism" in the sense of a Palestinian independence movement, like your question asks!

Pre-WWI Palestinism

This portion begins with an examination of where Palestinian identity really began to form, and where the word "Palestine" even comes from. One of the earliest, if not the earliest, recorded occurrences of the phrase is under the Roman empire, as Syria Palaestina. The Romans divided the area into Palaestina Prima and Palaestina Secunda, according to Gerber. This division, done after the revolt of Bar Kochba in 132 AD (where the area was regained by Jews for 3 years, before being lost again to the Romans) is where most people derive the name "Palestine" from. Previous derivations go back to the Philistea, a name given by Greek writers to the area where the Philistines lived, between Tel Aviv and Gaza.

The phrase, while existing as an administrative district under the Romans, appears to have gone out of style in government after the Arab empires went out of fashion themselves. As Gerber notes:

The term Jund Filastin or “the administrative region of Palestine” was current in Arab parlance from some time after the establishment of Muslim rule in the Fertile Crescent in the mid-seventh century until 1250.

After this, real administrative use of the term "Palestine" was not made until far later, hundreds of years forward in history. However, references to "Palestine" were still made, during the duration of the Ottoman Empire.

It's important to note that at no point did the groups of people living on the land seem to sociologically consider themselves "Palestinians" and try to rebel. The Bar Kochba revolt was done to re-establish Judea, one of two Jewish kingdoms of antiquity (the other being Israel). Palestinians did not appear to identify as "Palestinians" alone, and placed more root in their Islamic, or Arab, culture (or both).

During the Ottoman Empire, where our story picks up now, the area of "Palestine" as we understand it was divided into three, and separated from Syria. Harms explains:

Into the late-Ottoman years, Palestine was not a singular administrative geopolitical entity. Its organization changed over the course of the nineteenth century, but by the end of the century it was divided into three districts, or sanjaks: Jerusalem, Nablus, and Acre, all of which had been a part of the vilayet (governorate) of Syria.

In 1887, Jerusalem was made its own sanjak, and those of Nablus and Acre were transferred to the new vilayet of Beirut. So Jerusalem, which was a district not pertaining to any governorate, answered directly to the Ottoman central administration in Istanbul and had control over the south of Palestine as we know it. It is important, again, to note that the Ottomans frequently referred to the general area as Arz-i-Filistin (the "Land of Palestine") in correspondence.

Again, however, there is no real independence movement that came up in the early Ottoman period. Sociologically, however, the people of the area had already begun to call themselves "Palestinians" in a sense, as far as we can tell from cultural references. Mujir al-Din al-Hanbali al-Ulaymi, a resident of Jerusalem who wrote a history in 1490 of the area, used the term "Palestine" quite often in his work, and does so as a reference to the country he lives in. This is not to suggest that Palestine was a separate country, because it was not, but he understands the general area to have been Palestine, creating an inherently obvious link sociologically between how Palestinian identity considered itself back then. Khayr al-Din al-Ramli, who lived in al-Ramla in the mid-17th century and was a mufti and legal scholar, wrote Islamic legal discussions of questions posed by members of the public, called fatwas. A prominent writer, his works were cited quite often, as fatwas were public and supposed to be used by others. He, too, calls his country Palestine, and uses biladuna (our country) as well in his writings, suggesting that Palestine was definitely recognized as an area that people lived in, though not a governmental one.

Again, however, under the Ottomans there was not really a Palestinian independence movement. Typically the movement, at least pre-Mandate and pre-Zionism, rested on Arab nationalism. During the first World War, this Arab nationalism (pan-Arabism) was utilized by the British through Faysal Husayn to organize the Arab Revolt against the Ottoman Empire. The promises made to Husayn still cause a lot of problems today, and are the root of Arab grievances against the British as far as Palestine goes.

There is also a sentiment that Palestinian "states" had existed in the past. The best example of this is the principality in the area of the Galilee controlled for several decades until 1775. A family of Bedouin tax collectors called Zaydan built the principality under the Ottomans, under the ruling head Dahir al-Umar. In the 20th century, an anthropologist found that 20th century Palestinian peasants in the Galilee appear to have considered Dahir al-Umar as presiding over a Palestinian "state", but this was not independent of the Ottomans and the anthropologist's results have not been rigorously tested as we do today in most sociological studies.

The phrase Palestine, as far as modern usage, seems to have been brought up in an independence-movement sense by Filastin (Palestine, basically), a newly established newspaper founded by Isa al-Isa in 1911. Filastin was a prominent critic of the Ottoman government over how it perceived the Ottomans handling the Zionist movement.

Pre-WWI, the basic idea however was not to create a Palestinian state, but to create a pan-Arab one. This was the idea that led Faysal Husayn to organize the Arab Revolt against the Ottomans during WWI with the help of the British, and seems to have been the prominent ideology at the time among all others, at least before the war itself. Ottomanism and Islam were the prominent identity-builders, according to Khalidi, before WWI happened. However, Islam/Ottomanism stopped being the primary identifying factors early in the 20th century. As Khalidi puts it:

...the decline of religion as the governing principle of the Ottoman state in the waning years of the Empire, and what was perceived by many as the cynical exploitation of Islam by the highly secular CUP from 1908 to 1918, accelerated a decline in the saliency of religious identification in the Empire before and during the War. This complemented and enhanced a growing shift to secularism and secular nationalism on the part of some of the younger segments of the Ottoman elite, but this shift was by no means as definitive as the eclipse of Ottomanism.

The main identifying factor, then, became a vacuum. Over time, this vacuum was filled by both pan-Arabism (which died out quickly), and by Palestinism.

Khalil al-Sakakini, the co-founder of the al'Madrasa al-Wataniyya school in Jerusalem, said in 1925:

A nation which has long been in the depths of sleep only awakes if it is rudely shaken by events, and only arises little by little...This was the situation of Palestine, which for many centuries had been in the deepest sleep, until it was shaken by the great war, shocked by the Zionist movement, and violated by the illegal policy [of the British]; and it awoke, little by little.

With that, let's look into post-WWI Palestinism, and why/how it developed as it did.