Distance between two vertices is the number of edges in a shortest path connecting them in a connected graph G. The transmission of a vertex v is the sum of distances from v to all the other vertices of G. If transmissions of all vertices are mutually distinct, then G is a transmission irregular graph. It is known that almost no graphs are transmission irregular. Infinite families of transmission irregular trees and 2-connected graphs were presented in Alizadeh and Klavžar (2018) and Dobrynin (2019) [8, 9]. The following problem was posed in Alizadeh and Klavžar (2018): do there exist infinite families of regular transmission irregular graphs? In this paper, an infinite family of 3-connected cubic transmission irregular graphs is constructed.