There are camp sites all over the island but I don't know if they offer special long term rates.
The Polis campsite is right on the beach and only a short walk from restuarants/bars so a good option. Just contact them on 00357 26815080 for prices.
Helen
PS Sept/Oct is NOT really winter (Jan/Feb is when it's cold and rainy)- it's still hot and even November can be very warm - this year it averaged 26 degrees during the day.