Changes in serum copper concentration are observed in patients with depressive symptoms. Unmet needs in contemporary antidepressant treatment have increased interest in non-monoaminergic antidepressants, such as ketamine, an anaesthetic drug that has demonstrated a rapid antidepressant effect in patients with treatment-resistant depression (TRD). The purpose of this study was to examine whether serum copper concentrations change during ketamine treatment and whether there is an association between the copper concentrations and treatment response measured using psychometric scale scores. Moreover, the interlink between somatic comorbidities and copper concentration was studied. Patients with major depressive disorder or bipolar disorder were rated weekly by a clinician using the Montgomery–Asberg Depression Rating Scale (MADRS) and Young Mania Rating Scale (YMRS). Copper level assessments were carried out weekly before the start of ketamine treatment and then after every second infusion and one week after the last ketamine infusion. The serum concentration of copper before ketamine treatment was significantly higher than that after the fifth infusion (p = 0.016), and the serum concentration after the treatment was significantly higher than that after the fifth infusion (p = 0.048). No significant correlations between changes in the copper serum concentrations and MADRS or YMRS were found. The serum copper level was not associated with somatic comorbidities during the course of treatment. This study provides data on the role of copper in short-term intravenous ketamine treatment in TRD, although no clear evidence of a connection between the copper level and treatment response was found.