Is every local Gorenstein, integrally closed, integral domain, going to be an unique factorization domain?
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No. As a simple example, take $k[[x,y.z]]/(z^n-xy)$. – Mohan Oct 25 '19 at 18:13
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@Mohan Are there similar counter-examples if we additionally assume that the local ring corresponds to the localization at an isolated singularity of dimension at least $3$? – Chen Oct 25 '19 at 19:36
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Take $k[[x,y,z,w]]/(xy-zw)$. For larger dimension these exist, but not as easy as these by Grothendieck-Lefschetz. – Mohan Oct 25 '19 at 22:17