The book "Beginning Perl" by Simon Cozens and later on James Lee also contains some exercises. The first edition of the book is available online.

From the page: "Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve". One can learn a lot from solving these challenges. In case you're having a difficulty, then you can find solutions to some of these problems here:

1. Shlomi Fish's Solutions to Project Euler - mostly in Perl.

2. Solutions to the Euler Problems in the Haskell programming language - on the Haskell Wiki.

3. A web search for a "euler" and a particular question number may also provide some insights.

But remember to always try to solve the challenges on your own first.

A now inactive series of challenges for solving using Perl that was started by Mark Jason Dominus. One can go over the archives of the mailing lists for some interesting problems to solve and their solutions.

A list of 99 challenges with a focus on the Prolog programming language, but which can be solved using Perl.

A concentrated list of Project Ideas - on the Dream in Code forums.

Contains some challenges.

Contains a series of tasks, with solutions presented in as many programming languages as possible. Note: the Perl solutions presented there can vary in quality. (As can the solutions in other languages.)

A comprehensive list of sites with problem sets and/or for competitive programming - many of which can be solved in Perl.