In stepped-wedge trials (SWTs), the intervention is rolled out in a random order over more than 1 time-period. SWTs are often analysed using mixed-effects models that require strong assumptions and may be inappropriate when the number of clusters is small. We propose a non-parametric within-period method to analyse SWTs. This method estimates the intervention effect by comparing intervention and control conditions in a given period using cluster-level data corresponding to exposure. The within-period intervention effects are combined with an inverse-variance-weighted average, and permutation tests are used. We present an example and, using simulated data, compared the method to (1) a parametric cluster-level within-period method, (2) the most commonly used mixed-effects model, and (3) a more flexible mixed-effects model. We simulated scenarios where period effects were common to all clusters, and when they varied according to a distribution informed by routinely collected health data. The non-parametric within-period method provided unbiased intervention effect estimates with correct confidence-interval coverage for all scenarios. The parametric within-period method produced confidence intervals with low coverage for most scenarios. The mixed-effects models' confidence intervals had low coverage when period effects varied between clusters but had greater power than the non-parametric within-period method when period effects were common to all clusters. The non-parametric within-period method is a robust method for analysing SWT. The method could be used by trial statisticians who want to emphasise that the SWT is a randomised trial, in the common position of being uncertain about whether data will meet the assumptions necessary for mixed-effect models.