This question is answered for a large number of reductive groups (including the symplectic)  in P. Delorme and V. Sécherre, "An analogue of the Cartan decomposition for p-adic reductive symmetric spaces" ([arXiv](http://lanl.arxiv.org/abs/math/0612545), [MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2794612), [published](https://msp.org/pjm/2011/251-1/p01.xhtml)).