  We consider a generalisation of the Basilica group to all odd primes: the
$p$-Basilica groups acting on the $p$-adic tree. We show that the $p$-Basilica
groups have the $p$-congruence subgroup property but not the congruence
subgroup property nor the weak congruence subgroup property. This provides the
first examples of weakly branch groups with such properties. In addition, the
$p$-Basilica groups give the first examples of weakly branch, but not branch,
groups which are super strongly fractal. We compute the orders of the
congruence quotients of these groups, which enable us to determine the
Hausdorff dimensions of the $p$-Basilica groups. Lastly, we show that the
$p$-Basilica groups do not possess maximal subgroups of infinite index and that
they have infinitely many non-normal maximal subgroups.
