State law does not require that the members of a public facilities district (PFD) board of directors be city residents. RCW 35.57.010((3)(a), which addresses the board of directors of a single-city PFD, states in relevant part as follows:
A public facilities district created by a single city or town shall be governed by a board of directors consisting of five members selected as follows: (i) Two members appointed by the legislative authority of the city or town; and (ii) three members appointed by legislative authority based on recommendations from local organizations. The members appointed under (a)(i) of this subsection, shall not be members of the legislative authority of the city or town. The members appointed under (a)(ii) of this subsection, must be based on recommendations received from local organizations that may include, but are not limited to the local chamber of commerce, local economic development council, and local labor council.
As you can see, the quoted language does not impose any residency requirements.
However, the city ordinance creating the PFD may establish residency requirements, so you should check that ordinance, if you have not already done so.
(Link to this question)