In the regularly varying time series setting, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with lp-norm above a high threshold. Our main result derives new large deviation principles of extremal lp-blocks. We show an application to cluster inference to motivate our result, where we design consistent disjoint blocks estimators. We focus on inferring important indices in extreme value theory, e.g., the extremal index. Our approach motivates cluster inference based on extremal lp-blocks with p< ∞ rather than the classical one with p = ∞ where the bias is more difficult to control, as illustrated in simulations from an example. Our estimators also promote the use of large empirical quantiles from the lp-norm of blocks as threshold levels which eases implementation and facilitates comparison.