That sounds like a spa to me

We called them a spa or an aesthetics clinic.

In the United States, we call them a Spa.

Non-medical treatments are done in a spa. Medical treatments related to beauty, such as Botox, are done at a Med-Spa.

In the UK, a salon/spa would provide non-invasive/non-medical beauty treatments. Other aesthetic treatments, such as injections, might be provided at a (skin/dermatology/beauty/aesthetic) clinic. Regulations mean that qualified, medical professionals are keen to distinguish between their treatments and non-medical, however, there is some crossover between services. :-)