The match operator is not particularly well suited to extract this data since the data has two dimensions.

Due to the different structure of captures in Perl 6 regexes that doesn't hold true for Perl 6 anymore. Here's a Perl 6 solution that extracts all trailing dates with one regex match:

use v6;

my $str = 
'2009-02-02 06:12:57,500 dates processed: 2009-01-31, 2009-01-29, 2009
+-01-30
2009-02-18 06:03:47,713 dates processed: 2009-02-16, 2009-02-17
2009-02-19 05:58:29,138 dates processed: 2009-02-18
';

token date { \d**4 '-' \d**2 '-' \d ** 2 };
regex line { ^^ \N* 'processed:' \s* <date> [','\s* <date>]* \s*  \n }
+;

if $str ~~ m/ ^ <line>+ / {
    for $<line> -> $l {
        print "Dates in line $l";
        .say for $l<date>;
    }
} else {
    say "no match";
}
[download]

(tested on Rakudo).