Jacquet, PiateskiShapiro, and Shalika defined new vectors for generic representations of $GL(n,F)$, where $F$ is a nonarchimedean local field. I know that this notion has been extended to $GSp(4,F)$. Is there an extension to other $p$adic groups?

The theory of new vectors for ${\rm GSp}(4)$ has been written by Schmidt and Roberts : Local Newforms for GSp(4). Springer Lecture Note in Mathematics, vol. 1918 (2007) See also Schmidt's webpage : http://www2.math.ou.edu/~rschmidt/ The definition is trickier than in the case of ${\rm GL}(N)$ By the way : there is a mistake in Jacquet/PiateskiShapiro/Shalika. It was pointed out and fixed by Matringe : arXiv:1201.5506 Essential Whittaker functions for GL(n). Nadir Matringe. See also Jacquet's webpage. New vectors are also known for generic representations of reductive groups of small ranks (in fact of rank $1$) : ${\rm SL}(2)$, unitary groups. There is no general theory (except for spherical representations). 

