I do not want to leave the impression that I am ungrateful. I am just busy putting all the above good advice to use.
While not necessarily everything suggested is directly (or easily) applicable in this case, be assured that even the bits that are not so right now are not lost and will be remembered.

At the moment, the program is still leaking, but I haven't lost hope. I have just tried the easiest bits first. I can see the wisdom of some of the more drastic recommendations, but some pills are bitter to swallow.

Many thanks to all anyway.

While I am digging, can anyone make an educated at guess at the phenomenon whereby, when I stop the program (properly, by a caught signal), and keep watching memory, said memory continues to increase (significantly) for a while, then starts decreasing until the program finally stops ? I mean, for instance does perl have to copy stuff in order to be able to destroy it ? The answer may possibly give me a clue as to what is going on.