Implantable devices capable of targeted and reversible blocking of peripheral nerve activity may provide alternatives to opioids for treating pain. Local cooling represents an attractive means for on-demand elimination of pain signals, but traditional technologies are limited by rigid, bulky form factors; imprecise cooling; and requirements for extraction surgeries. Here, we introduce soft, bioresorbable, microfluidic devices that enable delivery of focused, minimally invasive cooling power at arbitrary depths in living tissues with real-time temperature feedback control. Construction with water-soluble, biocompatible materials leads to dissolution and bioresorption as a mechanism to eliminate unnecessary device load and risk to the patient without additional surgeries. Multiweek in vivo trials demonstrate the ability to rapidly and precisely cool peripheral nerves to provide local, on-demand analgesia in rat models for neuropathic pain.