If you want to capture to the end of the line, in /m mode, $ anchors at the end of a line. In /s mode, ^ matches at the beginning of the string and $ matches at the end. So maybe you did want /m. Of course, you can also just do something like these:

while (<DATA>) {
   my ($key, $value) = /(\S+) (.*)/ or next;    # or:
   my ($key, $value) = split;                   # or:
   (undef, $key, $value) = split(/\s+/, $_, 3);

   $hash{$key} = $value;                        # or:
   push(@{$hash{$key}}, $value);
}
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Untested, but you might get some ideas from that.