The fabled English longbow remained in use much later than is generally credited.

When troops were levied for the 1627 English Isle de Rhe expedition against France the Privy Council ordered that 24% of the infantry should be archers. In practice, the counties struggled to find that number of men who could still shoot a long bow, and I've yet to find any records to show what proportion of the army was actually made up of archers. However, French sources do mention English archers at the siege of St. Martin de Rhe, and they are mentioned in the correspondence of English commanders, so we can be sure that some made it across the Channel. This appears to be the last time archers made up any significant proportion of an English army on the continent.

During the English civil war at least four company-sized units (c. 100 men in theory) were raised, and at least two saw active service in the field up to 1644. Odd mentions of archers in the field continue until at least 1647, and Montrose's highland army also included archers, though it's difficult to say how many. The last company of archers raised by Parliament was ordered in November 1643 and saw service in the Western campaign the following year, but they were raised in response to a perceived threat of whole regiments of archers being raised by the Royalists. I have been unable to find any evidence of the Royalists actually raising archers in large numbers, but the rumours were certainly believed at the time.

Sources, bibliography, and more details HERE

The Janissary ortas of the Ottoman Empire rather aggressively adopted the matchlock and later flintlock musket, but there were a number of Janissary ortas that would continue to use the compound bow "well into the 17th century," according to Gabor Agoston in Encyclopedia of the Ottoman Empire. He doesn't go into much more detail than that, unfortunately. The Kapikulu formations didn't have a standardized weapon like many of its contemporaries in the 17th century; Ottoman tradition was that the Kapikulu slave-soldiers could fight with pretty much whatever they pleased, provided they were sufficiently skilled in the use of the weapon. Certain weapon types were much preferred over others, of course; the Sipahis loved their lances and, when it came to close combat, the Janissaries were fans of the kilij.

The Sipahis, as cavalry, had a longer tradition of archery than their rivals, the Janissaries. I'm not terribly sure at what point the compound bow was withdrawn from service there, but it seems to be later than it was for the Janissaries. I can't find a concrete date, unfortunately. It just seems to disappear from service sometime in the mid to late 17th century.

In Rhoads Murphey's account of the Sieges of Baghdad in the early 17th century (Ottoman Warfare, 1500-1700), he states that the Persian defenders dedicated roughly one fifth of their entire force to archers (compared to two fifths for musketeers). It's implied, though not stated, that this was the sort of garrison that Persia had in Baghdad when the Ottomans recaptured it in 1638.