Some children with proven intrauterine Zika virus (ZIKV) infection who were born asymptomatic subsequently manifested neurodevelopmental delays, pointing to impairment of development perinatally and postnatally. To model this, we infected postnatal day (P) 5–6 (equivalent to the perinatal period in humans) susceptible mice with a mammalian cell‐propagated ZIKV clinical isolate from the Brazilian outbreak in 2015. All infected mice appeared normal up to 4 days post‐intraperitoneal inoculation (dpi), but rapidly developed severe clinical signs at 5–6 dpi. All nervous tissue examined at 5/6 dpi appeared grossly normal. However, anti‐ZIKV positive cells were observed in the optic nerve, brain, and spinal cord; predominantly in white matter. Co‐labeling with cell type specific markers demonstrated oligodendrocytes and astrocytes support productive infection. Rarely, ZIKV positive neurons were observed. In spinal cord white matter, which we examined in detail, apoptotic cells were evident; the density of oligodendrocytes was significantly reduced; and there was localized microglial reactivity including expression of the NLRP3 inflammasome. Together, our observations demonstrate that a clinically relevant ZIKV isolate can directly impact oligodendrocytes. As primary oligodendrocyte cell death can lead later to secondary autoimmune demyelination, our observations may help explain neurodevelopmental delays in infants appearing asymptomatic at birth and commend lifetime surveillance.