You may want to consider the following simple solution:

#!/usr/bin/perl -w
use strict;

open A, '<export_gesamt.asc' 
  || die "Failed to open: $!\n";

my $line = '$';

do {
  if ( $line eq /^\$\s+$/ ) {
    my $file = <A>;
    open ( OUT ">$file" ) 
      || warn "Failed to create file '$file': $!\n";
    print "Writing $file\n";
  }
  else {
    print OUT $line;
  }
} while ( $line = <A> );
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