How deep does the frost have to go before it becomes permafrost?
Permafrost is permanently frozen ground. However, most definition qualify it as permafrost if it remains frozen for at least two years.

The speed at which a temperature change travels down the ground depends on the thermal conductivity, and it varies between soils. As a thumb rule, the temperature at 10 m depth is pretty much constant and equal to the mean yearly temperature at the surface. This means that 10m is roughly the distance the heat wave travels in a year.

That means that if you manage to freeze the ground below 10m the next summer won't be able to thaw it within a year no matter how hot it is, so it will stay frozen for at least one and a half years. How much longer it will stay frozen will depend on the exact temperatures and lengths of the winter that created the frost layer, the winters after that and the summers in between. Without more details it is impossible to give a number.

Said that, a single cold snap is unlikely to significantly affect the temperature very deep into the ground, you need an extended period of cold weather and many cold winters in a row to do that.

An exceptionally cold winter followed by a mild summer could create a layer of frost that survive two years, but its depth will depend of the actual temperature profile over that period.

Anyway, if that frost layer thaw is not going to be on the next summer, but much later, because the heat will take a long time to travel down there. More on these delay on this answer.