No, it should not be put in the dishwasher. The brown deposits are probably lime. It is worth decalcifying the appliance regularly (approx. once a week if the water is hard and used daily!) or using distilled water for operation.
The manufacturer recommends a descaling solution as follows: Citric acid solution (30g pure citric acid powder dissolved in
1l of water) or a vinegar solution (household vinegar with 5% acid content mixed with water in a ratio of 1:2).