The correct option is A did
The question "Did Anne wash the dishes?" contains the verb 'did'.
The reply "Yes she ___" must also be filled with the same verb, as 'did' implies that she did wash the dishes. 'Yes she did' forms an elliptical clause.
The reply is an elliptical sentence, that means that some words have been omitted and instead have been implied earlier.
Hence, the verb 'did' is correct here and option A is right.
Option B: 'Did wash' does not form an elliptical clause because 'wash' requires an object 'the dishes'. We need a verb that does not require an object, like 'did'.
Option C: 'Is washing' is in the present continuous tense whereas the question is in relation to the past.
Option D: 'Washed' is a simple past tense verb, but it isn't of the same form as in the question asked before 'did wash'. The reply must be coherent with the question that has been asked.
Hence options B, C and D are incorrect.