We study Gaussian random fields on certain Banach spaces and investigate conditions for their existence. Our results apply inter alia to spaces of Radon measures and Hölder functions. In the former case, we are able to define Gaussian white noise on the space of measures directly, avoiding, e.g., an embedding into a negative-order Sobolev space. In the latter case, we demonstrate how Hölder regularity of the samples is controlled by that of the covariance kernel and, thus, show a connection to the Theorem of Kolmogorov-Chentsov.