Note: untested code, written as I post.

open FH, "$file";
{
  local $/;
  $text = <FH>;
}
for($text =~ /<A NAME="([^"]+)">(.*?)(?!<A NAME=")/) {
  #file name is $1
  #text is $2
}
[download]

Note, I saved time and used .*?. See death to dot star for why not to do that. Also, look at unrolling the loop. I leave that as an exercise to you, because I can't just hand you everything now can I : )

The 15 year old, freshman programmer,
Stephen Rawls