A rule to live by ...
A rule never proven wrong ...

Fear is bad, if fear is directing your moves, your are going in the wrong direction.
I sense fear in your words

When it comes to programming, there is a very good advice that says, that you should learn a new language every now and then.
I add that you should you be competant in different type of languages.

• A system programming language: c or c++ (why not both)
• SQL
• XSLT & XML
• A functional programming language: Haskell or OCaml
• JavaScript, to create flexible browsers UI.
• Scripting languages: Perl, Ruby, Tcl (I love Tcl)

The problem is the market values experts (kinda, nowadays we may argue against this), and jumping from learning one language to another, is risky, you can waste you valueble time, and end up only scratching the surface of a dozen different languages (I did that).
So, just do you best and strike a good balance. But just never bet everything on one language.

Should you stick with Perl?
Sure, and learn other languages too.
Does Perl have a bright future?
If we can only take a peak into the future, but I think you are asking the wrong question. You should care more about your future, not Perl's future (not that we want to put to waste the skills we've learned, but ...). Learning Perl will definitely make you a better developer, regardless of anything, and that is good for your future.