## Does every index \$p\$ subgroup of \$SL(2,\mathbb{Z}_p)\$ contain \$\Gamma(p)\$?

Does every index $$p$$ subgroup of $$SL(2,\mathbb{Z}_p)$$ contain the principal congruence subgroup $$\Gamma(p)$$?

Equivalently, must it be the preimage of an index $$p$$ subgroup of $$SL(2,\mathbb{Z}/p\mathbb{Z})$$?