Probability: Theory and Examples, R. Durrett (2010), chapter 8.
A course in probability theory, K.L. Chung (2000).
From a review of Chung's book:
There is a substantial chapter on invariance theorems, including Donsker's invariance principle, the Doob-Donsker proof of the Kolmogorov-Smirnov theorem and finally a general invariance principle of Skorokhod, the proof of which is only sketched.
for Donsker's theorem in the context of more general metric spaces, see:
Convergence of Probability Measures, by P. Billingsley (1999)