Do I need a permit for basement finishing in Bisbee?

Most likely yes. Basement finishing usually requires a permit in Arizona. Confirm the specifics with Bisbee.

Finishing a basement into living space adds rooms, wiring, and often plumbing, which local codes in Arizona generally treat as permitted work. Egress, smoke alarm, and insulation rules come from the code your city, town, or county adopted. Arizona has no statewide residential building code. Cities and towns adopt their own codes, and a county board of supervisors may adopt a building code for zoned unincorporated areas and may exempt areas zoned rural or unclassified (ARS 11-861), so the permit rule depends on where the property sits. Arizona's small-job licensing exemption (ARS 32-1121, subsection A, paragraph 14) covers only casual or minor work under $1,000 in total and does not apply to any job that requires a local building permit, so a hired contractor on permitted work must hold a Registrar of Contractors license. This basement project adds a bedroom, a bathroom or wet bar, or new electrical circuits, or moves the furnace or water heater. Finishing work of this kind normally needs a building permit and the matching trade permits. A basement bedroom needs an emergency escape and rescue opening, such as an egress window or door, which is one of the most common reasons basement work fails inspection. Plan for framing, electrical and plumbing rough-in inspections before insulation and drywall go up.