New Carthage was established in Southern Spain. The Carthaginians used it as a base for years so that they could march to Rome since they couldn't cross the sea. Rome did own the seas at the time.
Is it possible that the Catheginians bred with the locals in Spain?
It is essentially a given, but that isn't saying much. The underlying premise of your question betrays an essential misunderstanding of ancestry, since if we are talking about ~2200 years ago, common ancestry should be talked about on continental scales. All Europeans share common ancestors going back only 1,000 years. Double that, and we are likely talking about wide swathes of common ancestry - that is to say that anyone alive in the greater European world (by which I would include North Africa and the Near East, regions which would have strong trade relationships with the European continent) 2,200 who had descendants is quite probable as an ancestor of everyone alive today. I've written about common decent before here, but you would likely be interested in this study as well.
So yes, Carthaginians would be included in the ancestry of modern Spaniards, but that is kind of meaningless as they would be included in the ancestry of almost anyone in Europe and the Mediterranean World.