  We construct an example of a bounded degree, nonamenable, unimodular random
rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an
example of a bounded degree, unimodular random rooted graph with $p_c<1$ but
with an infinite cluster at criticality. These examples show that two
well-known conjectures of Benjamini and Schramm are false when generalised from
transitive graphs to unimodular random rooted graphs.
