The correct option is
C She supported me during difficult times.
The word, 'support' in the given sentence refers to an abstract noun which means something that serves as a foundation or pillar of strength. The related verb is 'support', which means to serve as a foundation for or to uphold a person/cause by backing it up. 'Supported' is the past tense form of the word, 'support'. Hence, option C is the correct answer.
Option A: The word, 'supporting' is used as an adjective to describe someone/ something which 'supports. It is also the present continuous form of the word, support'. However, in the clause, 'She extended her supporting', it is used as a noun (and not a verb). Also, the word, 'supporting' cannot be used as a noun - it is grammatically incorrect. Hence, this option is incorrect.
Options B & D: These options are incorrect as the word, 'support' is used as a noun and not a verb in these sentences