I’ve been searching for an answer to this question everywhere I can think of, but I haven’t come across a firm answer.
This is the main page for the module: http://ldap.perl.org/
It links to documentation, but in my searches of the docs, I’ve found no mention of thread safety.
As of version 0.4001, apparently not. Notice this comment from the file LDAP/Message.pm:
So it would seem that the author, at least at some point in the past, was intending to add thread support, but admittedly had not yet.
I hope this helps.