Using foreach loops only:

my @results = ("");
foreach my $subarray (@array) {
    
    my @tmp_results = ();
    my @subarray = @{ $subarray };
    
    foreach my $tmp_result (@results) {
        foreach my $element (@subarray) {
            my $string = $tmp_result . $element;
            
            push @tmp_results, $string;
        }
    }
    @results = @tmp_results;
}
print join "\n", @results;
print "\n";
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The trick is in the overwritting of @results with @tmp_results at the end of the outer loop, as well as in initializing @results with a single empty list in order for concatenation to work further down.

This could probably be written with several map's, but it might become difficult to read.