Try to find out when Cypriot script started, and the answer you get differs wildly depending upon the source. In part, this reflects differing opinions about whether they consider the writing system on early artifacts to be Cypriot or not. There are artifacts from ~1500 BC which haven’t been deciphered — in part because it probably is a different language, but partly because the Cypriot script evolved over time. (This early version of Cypriot script is sometimes called “Cypro-Minoan”.) The earliest Cypriot script writings that can be read are from around 800 BC in Cypriot Greek. Those who see it as one script then place the origin of Cypriot at 1500 BC; those who see it as two scripts place the origin at 800 BC.
The current expert opinion is that Linear B and Cypriot both descended from Linear A.
Like Byblos script, Cypriot used a small vertical mark as to separate words. However, the dividers don’t always appear where we would expect them. Sometimes particles and other small bits would be lumped in with nouns; sometimes even verbs and nouns would not be separated. We cannot know if they thought of those groupings as being “one word”, if they had some different idea of what that grouping meant, or if they just goofed.
We also cannot know how the concept of “a word” has been affected over time by use of word dividers. We English-speakers tend to think that a word is “something surrounded by spaces”, so “beekeeper” is one word, not two. If we didn’t have spaces, how would we know what “one word” was? (Also, in English, it is relatively common for a two-word phrase to turn into a hyphenated phrase, and then later to lose its hyphen. For example, “ear splitting” to “ear-splitting” to “earsplitting”.) Thus, I suggest that the concept of “one word” is fuzzy even for people in a highly literate culture with a 1200 year old tradition of separating words with spaces.
Some linguists believe that children do not learn words, but rather whole phrases. It might be that before humans got around to seeing dividers between words on a regular basis, that they would not have had the concept of “a word” so clearly.
Cypriot is also interesting for being (as near as I can tell) the first writing system to not use logograms. It is possible that Byblos script does not use logograms, but it has about 100 characters in it, which is a bit large for a pure syllabary.