Thanks so much for this input, it helps a lot. I have added it to the test report

There is the same problem with SignedRankTest.

By default, MannWhitneyTest uses asymptotic statistics, which is not appropriate for such small samples. Using instead the method "Permutations", with a reasonable number of permutations, (50, 100), I obtained zero as a Pvalue in both cases.

Is there any way to report this bug? It is extremely influential. I can't imagine how many datasets from animal studies I've processed with M-W in Mathematica in last 5 years... It will be a nightmare to revisit all the data once again, taking into account that a lot of data is already published. I'm shocked.

Thank you very much. One more request: as long as you will fix the MannWhitneyTest function, it would be great also to fix the behavior of Method->"Automatic". Now it always calculates p-value from asymptotic normal distribution for U. At the same time for small samples (<20) an exact p-value could be calculated based on permutations (not Monte Carlo, a direct formula/table instead). For instance, in the reported case of data1, data2 the exact p-value is the probability of getting 2 extreme cases: all ranks of sample 1 below the sample 2 and all ranks of sample 1 above the sample 2, which is exactly 2(6!6!)/(6+6)! = 1/462 ~ 0.00216.The asymptotic normal approximation for U gives 0.003053. This difference is not a big deal if a single M-W test is performed as both results are significant. But if multiple comparisons are required, then multiple M-W tests are applied with subsequent correction of p-values by for instance Holm–Bonferroni method. And here the difference between 0.002 and 0.003 might become important.