Time-to-event analyses are often plagued by both-possibly unmeasured-confounding and competing risks. To deal with the former, the use of instrumental variables (IVs) for effect estimation is rapidly gaining ground. We show how to make use of such variables in competing risk analyses. In particular, we show how to infer the effect of an arbitrary exposure on cause-specific hazard functions under a semi-parametric model that imposes relatively weak restrictions on the observed data distribution. The proposed approach is flexible accommodating exposures and IVs of arbitrary type, and enabling covariate adjustment. It makes use of closed-form estimators that can be recursively calculated, and is shown to perform well in simulation studies. We also demonstrate its use in an application on the effect of mammography screening on the risk of dying from breast cancer.