Anything in the $n-1$st term of the lower central series of the pure braid group should work. 
The pure braid group is generated by generators $\beta_{i,j}$ where the $i$th strand pushes a finger over the intervening strands and hooks with the $j$th strand.
Then, when $n=3$, the commutator $[\beta_{1,2},\beta_{1,3}]$ is Brunnian in in your sense.   For $n=4$ you can consider $[\beta_{1,2},[\beta_{1,3},\beta_{1,4}]]$, etc.

Something like $[\sigma_1^2,[\sigma_2^2,\sigma_3^2]]$ would also work.

The reason this works is that deleting a strand from the braid kills at least one generator involved in the iterated commutator, so that it collapses to $1$. That's why you need to include a generator $\beta_{i,j}$ that involves each strand.

(This has been edited to remove innacuracies of previous versions.)