No, what you’d need is more continuous derivatives between elements, not higher polynomial degree within each element. The Lagrange basis (of any polynomial degree) is still only C^0, i.e., derivatives up to 0^{\text{th}}-order are continuous. (You might take a look at my answer here for more discussion of inter-element continuity and variational formulations, in the simpler context of the Poisson equation.) The spline spaces from IGA allow you to control the number of continuous derivatives between elements. You would need at least C^1 continuity to discretize your PDE using a standard Bubnov–Galerkin method.
P.S. Another possible direction for higher-order PDEs is to use a discontinuous Galerkin method, such as the one used in the biharmonic tutorial, where there is a single C^0 field, but extra terms are added to the variational form, integrated on interior mesh facets. However, this is probably the most difficult to formulate.