Propensity score trimming, which discards subjects with propensity scores below a threshold, is a common way to address positivity violations that complicate causal effect estimation. However, most works on trimming assume treatment is discrete and models for the outcome regression and propensity score are parametric. This work proposes nonparametric estimators for trimmed average causal effects in the case of continuous treatments based on efficient influence functions. For continuous treatments, an efficient influence function for a trimmed causal effect does not exist, due to a lack of pathwise differentiability induced by trimming and a continuous treatment. Thus, we target a smoothed version of the trimmed causal effect for which an efficient influence function exists. Our resulting estimators exhibit doubly-robust style guarantees, with error involving products or squares of errors for the outcome regression and propensity score, which allows for valid inference even when nonparametric models are used. Our results allow the trimming threshold to be fixed or defined as a quantile of the propensity score, such that confidence intervals incorporate uncertainty involved in threshold estimation. These findings are validated via simulation and an application, thereby showing how to efficiently-but-flexibly estimate trimmed causal effects with continuous treatments.