https://en.m.wikipedia.org/wiki/Mass_media_in_Lebanon#Press
The link actually says its the first publication in the middle east.
http://en.wikipedia.org/wiki/Global_spread_of_the_printing_p...
Even in Lebanon the presses were introduced by Maronites; the Islamic authorities were apparently not fans of the technology. You will also notice the presses in Safed from 1577; it's likely that your 1610 example is referring to the first book, specifically.
Just for comparison, the first newspaper in Mexico started in 1539 (Mexico City), and the first in India in 1556 (Goa). The first book in the Ottoman Empire was printed in Constantinople in 1493. The Ottomans, then the largest empire in the world (until the Spanish conquest of Mexico), absolutely could have adopted printing a century before your date if they had just... wanted to. The list gives 1706 for the first continuously operated Arabic press in the Ottoman Empire at Aleppo.
I just felt the need to comment since I remember learning that factoid in my history class growing up and never had a chance to say it :)
This enormous act of self-sabotage probably kept the alphabets in the Muslim world from standardizing during the 1500s-1800s as the Europeans had, so variations in writing remained common. Pre-1400s texts from Europe look weird, too.