Does a current carrying wire produce electric field outside?

Outside a current carrying conductor, there is, in fact, an electric field. This is discussed for example, in "Surface charges on circuit wires and resistors play three roles" by J. D. Jackson, in American Journal of Physics – July 1996 – Volume 64, Issue 7, pp. 855.

To quote Norris W. Preyer quoting Jackson:

Jackson describes the three roles of surface charges in circuits:

  1. to maintain the potential around the circuit,
  2. to provide the electric field in the space around the circuit,
  3. and to assure the confined flow of current.

Experimental verification was provided by Jefimenko several decades ago. A modern experimental demonstration is provided by Rebecca Jacobs, Alex de Salazar, and Antonio Nassar, in their article "New experimental method of visualizing the electric field due to surface charges on circuit elements", in American Journal of Physics – December 2010 – Volume 78, Issue 12, pp. 1432.