  The isogeometric approximation of the Stokes problem in a trimmed domain is
studied. This setting is characterized by an underlying mesh unfitted with the
boundary of the physical domain making the imposition of the essential boundary
conditions a challenging problem. A very popular strategy is to rely on the
so-called Nitsche method [22]. We show with numerically examples that in some
degenerate trimmed domain configurations there is a lack of stability of the
formulation, potentially polluting the computed solutions. After extending the
stabilization procedure of [17] to incompressible flow problems, we
theoretically prove that, combined with the Raviart-Thomas isogeometric
element, we are able to recover the well-posedness of the formulation and,
consequently, optimal a priori error estimates. Numerical results corroborating
the theory and extending it for the case of the isogeometric N\'ed\'elec and
Taylor-Hood elements are provided.
