I wrote this answer a couple years ago, which addresses this question. The short answer is yes, and depending on how big you believe a rock has to be to be called an "island", there are still "islands" out there being found today.
Although satellites catch new very small ones, a new volcanic island in the North Atlantic or somewhere, the short answer for bigger land masses would be, no. That had been sorted out quite a long time beforehand. For the first several hundred years of European exploration there had been , to the contrary, quite a number of anomalous sightings of islands- a lookout on the dawn watch sees what he thinks is a shoreline at the foggy horizon, and a note is made in the log but the ship doesn't get any closer, or (because accurate means of measuring longitude happened only in the mid 18th c.), an island is seen in two locations on two different voyages by two different captains, and given two different names. This resulted in the listing of quite a number of islands that were not there. But as recordkeeping and navigation improved, ships that were blown off course could definitively say that they were at a location, and there was really no island there. And of course, if there was an island placed on a chart, and they thought they were being blown into it, they would naturally make a note on the chart that it didn't exist. Those imaginary islands slowly got crossed off the charts,
Some new ones were discovered rather late, in the 19th c.. A very important part of this discovery process were the 19th c. whalers. Unlike most ships, they did not follow routes from port to port but went out looking for whales, and stayed out for years at a time, criss-crossing even the lowest and highest latitudes in the hunt, and making notes in their logs. So, a British whaler ( or, more precisely , a sealer) would discover Deception Island ( 38°43'14.9"S 77°30'34.5"E) in 1820 in the Antarctic because no one before had had any reason to be in the area. Those whalers also made notes of where they DIDN'T see anything. After decades of whaling, all the blank spots of the oceans had been crossed and described, all long before satellites were in orbit.
Henry Stommel: Lost Islands: the Story of Islands that have Vanished from Nautical Charts
EDIT: I was thinking of islands as places that would hold a Gilligan for a day or two, should have realized tiny ones do appear and do get tagged by satellite. I have amended.