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A bit further, around 60km from Beer-Sheva, between Ashqelon and
Ashdod, few kilometers from Kibbutz Nizzanim (reference in Hebrew) there is a nice Nizzanim beach. It seems to be the closest one from Beer-Sheva beyond the reach of Qassamim rockets, which are on daily basis fired from Gaza Strip towards Sderot and occasionally towards Ashqelon.
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Camping is accepted in all above mentioned places. There are toilets and showers, but they may be closed during winter. If you come by car, then in Zikim and Nizzanim you may be asked to pay an entrance fee. Seemingly, the charge does not apply to cyclists and motorcyclists.
Three photos below were taken in Nizzanim in the end of December.
