Is there a non-blocking operation to back up MyISAM tables using Percona XtraBackup?
No - the MyISAM architecture prohibits hot backups - it uses a locking mechanism. You might be interested to my response in a related thread.

Take a mysqldump of your database, change the bit where it says ENGINE=MyISAM to ENGINE=InnoDB (InnoDB uses a MVCC architecture) and then restore - XtraBackup will then perform hot backups. It's relatively easy to use sed to do the edit that I have suggested 

mysqldump [options] | sed 's/ENGINE=MyISAM/ENGINE=InnoDB/g' &gt; my_schema.sql 


Then restore your dumpfile in the normal way.

MyISAM is going to be deprecated in the near future anyway! [EDIT - added 19/04/2018] - the system tables in MySQL 8 are now InnoDB!

In reply to the OP's question about the Percona's XtraBackup page being a "complete fabrication" -  the quote is "can back up the following storage engines by briefly pausing writes at the end of the backup".

I think that the keyword on that page is  "briefly"! :-) Take a look here - Bill Karwin is a big hitter on this forum.

Also, take a look at what Shlomi Noach says about MyISAM hot backups (or lack of!) - he's also a big hitter in the MySQL world. The file system snapshot to which he refers is mylvmbackup - which appears to have a reasonably good rep. here.