I have a problem with beta diversity analysis - looking at the obtained graphs, you can see large differences between the groups, but the p-value is high and the results are statistically insignificant. Is it even possible? Have I done something wrong?
I see a large overall difference between groups. However, as you get into the pairwise comparisons with penalties, your sample sizes are small and you're paying a high penalty for multiple hypothesis correction.
At 6 samples (3 per group) your permutations pace is limited to 6! = 720, which is actually less than the 999 permutations you specified in our overall model.
Very few techniques will work with fewer than 5 samples per group. Permanova (beta diversity) kruskal wallis (alpha diversity), and differential abundance with ANCOM are all based on tests that are recommended with at least 5 samples per group (more is ideal).