Your chairs may have mildew or mold in them. Wicker is an organic material and will pick up odors from the environment. I would suggest cleaning your wicker chairs with a solution of 10% bleach in water. Use 1 part bleach to 9 parts water. (Mildew usually appears as small black dots.) Be sure to rinse well and let your wicker furniture dry thoroughly. This may take several days.
After the wicker is completely dry, additional protection can be provided for your wicker by applying a finish. Varnish, shellac, or lacquer can be used. After the finish is completely dry, add a thin coating of liquid furniture wax to maintain a glossy appearance.

wicker itself doesn't smell, but it does absorb odors. Since you washed it, the water accentuates the smell. You have to let it dry completely in an open air environment.

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All of the above ideas are good. Also let the chairs dry in the sun if you can. The sunshine can do wonders. Good luck