Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites

We consider the i.i.d. Bernoulli field $\mu_p$ on $\mathbb{Z}^d$ with occupation density $p\in [0,1]$. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large $p$, as it changes only a small fraction $p(1-p)^{2d}$ of sites, there is $p(d)<1$ such that for all $p\in(p(d),1)$ the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small $p$, the Gibbs property is preserved.