Yes, you can!
For the Boxes, Wood Pieces and Balloons:
All the pieces can be cleaned with a damp cloth. Make sure to dry them with some napkins or a dry cloth.
For the Playmat:
If you accidentally spill liquids or water on the Playmat, towel dry it carefully, thoroughly, and quickly to ensure no damage occurs. Pay special attention to the edges of the mat, which are more vulnerable to water damage.
Note that any kind of dry erase, wet erase, or other markers will permanently stain and ruin the mat.