use vars would do the trick. I'd probably use our.

sub p1 {
    our $persistant;  $persistant = "init" unless defined $persistant;
   
    $persistant .= ".";

    return $persistant.
}
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Of course, that's going to be global to the package namespace, so if you need more than one type of p1(), maybe bless can help too.

package niffty;

sub new { bless { persistant => "init" } }
sub p1 { my $this = shift; $this->{persistant} .= "."; 
    return $this->[persistant}; }
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