How about Australia and Tasmania? The indigenous languages on each side of the Bass Strait have at least one feature in common (initial Cr- clusters) that's rare elsewhere in Australia. (There might be more, but information is pretty hard to come by, and the Tasmanian languages are sparsely attested.) That may or may not be indicative of continued contact; it might even be a coincidence, but if it's not, I wouldn't expect the shared feature to stick around for thousands of years.
(If Wikipedia is right about the time of Tasmania's separation from the Australian mainland and Tasmania was already populated at the time, the feature would've had to have been preserved on both sides of the Bass Strait for about twice the time between modern English and Proto-Indo-European.)
(If Wikipedia is right about the time of Tasmania's separation from the Australian mainland and Tasmania was already populated at the time, the feature would've had to have been preserved on both sides of the Bass Strait for about twice the time between modern English and Proto-Indo-European.)