- Context. The Cluster Lensing And Supernovae survey with Hubble (CLASH) is a Hubble Space Telescope (HST) Multi-Cycle Treasury programme that observes 25 massive galaxy clusters, 20 of which were X-ray-selected to preferably choose dynamically relaxed clusters, and 5 additional “high magnification” clusters, which were selected based on their optical lensing properties. CLASH aims to study the dark matter distribution of the clusters and find magnified high-redshift galaxies behind them. CLASH observations were carried out in 16 bands from UV to NIR to derive accurate and reliable estimates of photometric redshifts. Aims. We present the CLASH photometric redshifts using 16 HST bands and study the photometric redshift accuracy including a detailed comparison between photometric and spectroscopic redshifts for the strong lensing arcs using the measurements from the cluster MACSJ1206.2-0847. Methods. We used the publicly available Le Phare and BPZ photometric redshift estimation codes on 17 CLASH galaxy clusters for which the full photo-z data processing had been completed at the time of this analysis, and derive an estimate of the CLASH photo-z accuracy. Results. Using Le Phare code for objects with a S/N ≥ 10, we reach a precision of 3%(1 + z) for the strong lensing arcs, which is reduced to 2.4%(1 + z) after removing outliers. For galaxies in the cluster field, the corresponding values are 4%(1 + z) and 3%(1 + z). Using mock galaxy catalogues, we show that 3%(1 + z) precision is what is expected using the baseline sky substraction algorithm when taking into account extinction from dust, emission lines, and the finite range of SEDs included in the photo-z template library. An improved method for estimating galaxy colours that yields more accurate photometric redshifts will be explored in a forthcoming paper. We study photo-z results for different aperture photometry techniques and find that the SExtractor isophotal photometry works best. We check the robustness of the arcs photo-z results by rederiving the input photometry in the case of MACS1206. We describe and release a photometric redshift catalogue of the MACS1206 cluster we study. Conclusions. Our photo-z codes give similar results for the strong lensing arcs, as well as for galaxies of the cluster field. Results are improved when optimizing the photometric aperture shape that shows an optimal aperture size around 1′′ radius, giving results that are equivalent to isophotal photometry. Tailored photometry of the arcs improves the photo-z results by showing more consistency between the different arcs of the same strong lensing system.