So, I can use $p::var as some arbitrary global variable even under "use strict" and even when "p" is not referring to any package at all?

In that example, "p" is is referring to a package. It doesn't matter that you didn't enter the package namespace anywhere with pacakge p; first though. Yes, you can use it even under use strict; because it is fully qualified. Even strict checking assumes that you know what you are doing when you fully qualify a variable name.

Is it an advisable thing to do?

Use global variables from random packages? No. And it is probably better to predeclare globals with use vars (or our which isn't exactly the same thing) than to not declare and use fully qualified names.

-sauoq
"My two cents aren't worth a dime.";