Untested, but perhaps something such as:

LOOP: while (my $line = <DATA>) {
    my $line2 = undef;
    if ( $line =~ m/^\s+/ ) {
        while (my $line2 = <DATA>) {
            if ( $line2 =~ m/^\s+/ ) {
                $line .= $line2;
            }
            else {
                last;
            }
        }
    }

# ... remaining processing here ...

    if (defined $line2) {
        $line = $line2;
        redo LOOP;
    }
}
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Hope that helps.