The answer is no

In this dialogue, I think it is clear that the woman said there is no cooking oil left!

I've consulted dictionaries to find "not that I know of" a veru confusing phrase. It has at least 2 meanings: "so far as I know, the fact is not like this" and "so far as I know, there is nothing."

Could you think of other examples of its usage?

In response to a question, 'Not that I know of' simply means 'No, as far as I know'. The person answering is being a little cautious as his/her knowledge may not be complete. There may be a bottle of cooking oil hidden in a cupboard.