Is there a known minimal stretch of DNA that can distinguish any two people in the world?
Here is what the data says. UK government must have had some scientific evidence when it settled on a 10 variable-length sections of genome for their database, SGM+. In one such variable sections, some people have 10 repeats of CTTT, others have 11, others have 12 etc. The largest of those fragments, at its maximum length, are about 350 base pairs. The US government uses 13 such variable-length sequences for its CIA database, called CODIS. On both sides of the Atlantic, roughly 3-4 thousand base pairs were judged to be enough, with some safety margin, by the most advanced geneticists in the world, working on public money.

The catch is that these variable length sequences are disjoint, so you end up requiring far more than 4,000 bp. If you need these markers on the same DNA strand (chromosome), you should space them by 50 centimorgans, at which distance they recombine independently. But according to http://www.sciencemag.org/site/feature/data/genomes/265-5181-2094.pdf , the longest chromosome only has 350-400 centimorgan. It means you cannot get 10 independent DNA sequences on a human chromosome, be it the longest.

I predict there is no such continuous DNA stretch, that would tell apart any two non-twins. Since it is much better to use multiple chromosomes in DNA fingerprinting, I doubt you will find more relevant experimental data.