die is not the way to normally end execution. And for can iterate a list without indexing.

my $posn = -1;
my $same = 1;
while(defined $same and $posn <= length $files[0]){
 my $chr = substr($files[0], ++$posn, 1);
 for my $name (@files) {  
   undef $same and last if(substr($name, $posn, 1) ne $chr
                                 or length $name < $posn);
 }
}

print 'Prefix is "', substr($files[0], 0, $posn), '"';
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# UNTESTED

$max = ( sort { $a <=> $b } map { length } @files )[0];
# length of smallest filename - I hope.

LIST:
for $len ( 1..$max) )
{
    $s = substr($file[0],0,$len);
    last LIST unless scalar grep { /^$s/ } @files == @files;
    # precedence issue here?
}

print "prefix: '",substr($file[0],1,$len-1), "'\n";
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Update: added line to get min length of any filename.