  We construct Grassmannian categories of infinite rank, providing an infinite
analogue of the Grassmannian cluster categories introduced by Jensen, King, and
Su. Each Grassmannian category of infinite rank is given as the category of
graded maximal Cohen-Macaulay modules over a certain hypersurface singularity.
We show that generically free modules of rank $1$ in a Grassmannian category of
infinite rank are in bijection with the Pl\"ucker coordinates in an appropriate
Grassmannian cluster algebra of infinite rank. Moreover, this bijection is
structure preserving, as it relates rigidity in the category to compatibility
of Pl\"ucker coordinates. Along the way, we develop a combinatorial formula to
compute the dimension of the $\mathrm{Ext}^1$-spaces between any two
generically free modules of rank $1$ in the Grassmannian category of infinite
rank.
