The nearest points in a set
This problem does generalize the clique problem.

One example of a metric is the shortest path distance in a connected, unweighted graph $G$, where the distance between vertices $u$ and $v$, $d(u,v)$, is the length of the shortest path between them. There exists a clique of size $K$ in this graph if and only if we can find $K$ points whose distances sum to ${K \choose 2}$.

For finding cliques of size $5$ you can do slightly better than brute force: see this answer. However without more structure on the metric you are interested in it is an open problem to do significantly better.