From isotope dating of the Greenland and Antarctic (Vostok and Antarctic Plateau sites) ice cores, it is pretty well evidentially established that the last ice age started to end about 13,000 years ago and finished 11,700 years ago - which marks the start of the Holocene.
There's evidence that mammoths were still around in Siberia just under 10,000 years ago, and isolated populations survived on islands north of the Aleutians as few as 5700 years ago.
There's no evidence of mammoths surviving on the European or North American mainlands out to the Little Ice Age, which was just 500-150 years ago.
There may be evidence of residual mammoth populations on other isolated islands, but at the moment there is no positive evidence for your hypothesis.
Scientific theories are always provisional, and open to being overturned by new evidence. Nothing new there.