From what I can observe by looking at Service Design job posts, yes, there are possibilities to work part time as a service designer. But there aren’t a lot of job posts that clearly advertise it. Such positions are rather the exception. As proof of that, just do a search on all job posts from the website servicedesignjobs.com that contain the term "part-time", and you won't find many.
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Nevertheless, even when seeing a full-time job offer, you can always try to negotiate to make it a part-time one.
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And if that doesn’t work out, there is still the route of being a freelance service designer. The market for freelance service designers doesn't have the same maturity level in every country. For example, the demand is pretty good if you are freelancing in London but might be pretty low if you are freelancing in smaller cities.