Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is allowed.

Are there second (or even higher) order infinitary languages? I mean something like $\mathcal L^2_{\kappa, \lambda}$ (more generally $\mathcal L^n_{\kappa, \lambda}$)rendering of the infinitary first order language $\mathcal L_{\kappa, \lambda}$.

If there are, what are the recommended sources on those?

Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is allowed.

Are there second (or even higher) order infinitary languages? I mean something like $\mathcal L^2_{\kappa, \lambda}$ (more generally $\mathcal L^n_{\kappa, \lambda}$)rendering of the infinitary first order language $\mathcal L_{\kappa, \lambda}$. Or, is it the case that all of those are reducible to first order infinitary languages, and so dispense with all of them?

If there are, what are the recommended sources on those?