Your main problem is here:

while (my $conn = $socket->accept) {
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This doesn't happen in somewhat older perls since they by default have a setting to restart slow systemcalls (but the problem would reappear in really old perls which didn't yet do that).

In new perls (since 5.8) signals only set a flag and handlers only get executed if the perl dispatcher is in a safe state and sees the flag. But that forced the developers to turn off the restarting of slow systemcalls since otherwise actual running of the signal handler code can be indefinitely delayed (the flag would get set, but since the systemcall gets restarted it still doesn't get to the dispatcher, so nothing gets executed)

On UNIX versions that support it, you could avoid getting child signals at all by using

$SIG{CHLD} = "IGNORE";
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So just change the loop to something like:

use POSIX qw(EINTR);
...
while (1) {
   my $conn = $socket->accept;
   if (!$conn) {
       warn("Accept error: $!" if $! != EINTR;
       next;
   }
   ....
}
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Thanks to everybody that replied. This seems to be working as I want. I will read more about the EINTR signal. When the last child dies the error rasised by $socket->accept is "No child processes". I will look at Net::Server. It looks like what I need, but I am glad I took this approach to start. Again, my many thanks.

Ian