  Given a branching random walk on a set $X$, we study its extinction
probability vectors $\mathbf q(\cdot,A)$. Their components are the probability
that the process goes extinct in a fixed $A\subseteq X$, when starting from a
vertex $x\in X$. The set of extinction probability vectors (obtained letting
$A$ vary among all subsets of $X$) is a subset of the set of the fixed points
of the generating function of the branching random walk. In particular here we
are interested in the cardinality of the set of extinction probability vectors.
We prove results which allow to understand whether the probability of
extinction in a set $A$ is different from the one of extinction in another set
$B$. In many cases there are only two possible extinction probability vectors
and so far, in more complicated examples, only a finite number of distinct
extinction probability vectors had been explicitly found. Whether a branching
random walk could have an infinite number of distinct extinction probability
vectors was not known. We apply our results to construct examples of branching
random walks with uncountably many distinct extinction probability vectors.
