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I don't have an actual answer for you, but I'll give you some possibly helpful buzzwords to try to google your way to an answer: stable random vectors. As I understand it (at least in the case $p=2$), the existence of 1-stable random vectors is essentially the positive definiteness of $\exp(-\|x\|)$. So you want to know if a $1$-stable random vector in $\Bbb R^n$ has a strictly positive density. There are probably probabilists who know this, but I'm not one of them and don't have the time to hunt it down right now.