We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call $p$-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with $p$-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in van der Vaart and van Zanten 2008, namely we use prior concentration to verify prior mass and entropy conditions sufficient for posterior contraction. However, the specific concentration properties of $p$-exponential priors lead to a more complex entropy bound which can influence negatively the obtained rate of contraction, depending on the topology of the parameter space. Subject to the more complex entropy bound, we show that the rate of contraction depends on the position of the true parameter relative to a certain Banach space associated to $p$-exponential measures and on the small ball probabilities of these measures. For example, we apply our theory in the white noise model under Besov regularity of the truth and obtain minimax rates of contraction using (rescaled) $α$-regular $p$-exponential priors. In particular, our results suggest that when interested in spatially inhomogeneous unknown functions, in terms of posterior contraction, it is preferable to use Laplace rather than Gaussian priors.