This paper is mainly concerned with extensions of the so-called Vishik functional calculus for analytic bounded linear operators to a class of unbounded linear operators on $c_0$. For that, our first task consists of introducing a new class of linear operators denoted $W\left(c_0(J,\omega ,𝕂)\right)$ and next we make extensive use of such a new class along with the concept of convergence in the sense of resolvents to construct a functional calculus for a large class of unbounded linear operators.