I live in the frozen North. Frosts (of any kind light or hard) and tender plants like basil, tomatoes, etc. do not mix. Bring in or cover (only if the frost is to be light) any tender plants if there are frost warnings. If only the tips or tops of a tender plant was frosted, it can survive. But annuals generally give up the ghost if frosted too hard. But you can protect your "mother" plant from additional frosts and see how it does.

FYI - In my experience, basil does not do well when the temperatures drop below 50 F on a regular basis. It stops any vigorous growth and begins to get rather sick looking. I know. I get to see this each fall.

FataMorgana

I live in a warm part of California, but never plan on having basil survive even the slightest cold. Besides that, IMHO, old basil plants don't provide the tastiest leaves.

Okay. I won't keep my hopes up. But since I don't have immediate plans for this spot, I will leave the mother plant there. If it comes back... Yay. If it doesn't come back... Yay. Free compost material.

On a side note: Even cutting the dead stems of the basil resulted in a very pleasant scent in the garden. I found myself not just cutting the stems from the roots but repeatedly cutting the stems and letting them drop into the mulch. For no more reason than to release the scent.

It probably won't come back. I'm in 9a, as well, but Texas, and we had 1 night that went down to about 29, and the basil is history. I plant lemon basil - it makes a great tea, and reseeds itself every spring, unlike other basils.