How about an alternation in the look-ahead? I also made the capture non-greedy.

johngg@shiraz:~/perl/Monks > perl -Mstrict -Mwarnings -E '
open my $cmdFH, q{<}, \ <<EOD or die $!;
BSS <blah blah>
   SSID: <ssid>
   <blah blah>
BSS <blah blah>
   SSID: <ssid2>
   <blah blah>
BSS <blah blah>
   SSID: <ssid3>
   <blah blah>
EOD

my $out = do {
   local $/;
   <$cmdFH>;
   };

my @ssids = $out =~ m{SSID: (.*?)(?=(?:^BSS|\z))}smg;
say qq{--<$_>--} for @ssids;'
--<<ssid>
   <blah blah>
>--
--<<ssid2>
   <blah blah>
>--
--<<ssid3>
   <blah blah>
>--
[download]

I hope this helps.