Context. Precision measurements of the anisotropy of the cosmic microwave background (CMB) are able to detect low-level nonGaussian features caused by either topological defects or the inflation process. These measurements are becoming feasable with the development of large arrays of ultra-sensitive bolometric detectors and their use in balloon-borne or satellite missions. However, the space environment includes a population of cosmic rays (CRs), which produce spurious spikes in bolometric signals. Aims. We analyze the effect of CRs on the measurement of CMB anisotropy maps and the estimate of cosmological non-Gaussianity and angular power spectra of the CMB. Methods. Using accurate simulations of noise and CR events in bolometric detectors, and de-spiking techniques, we produce simulated measured maps and analyze the Gaussianity and power spectrum of the maps for different levels and rates of CR events. Results. We find that a de-spiking technique based on outlier removal in the detector signals contributing to the same sky pixel is effective in removing CR events larger than the noise. However, low level events hidden in the noise produce a positive shift of the average power signal measured by a bolometer, and increase its variance. If the number of hits per pixel is large enough, the data distribution for each sky pixel is approximately Gaussian, but the skewness and the kurtosis of the temperatures of the pixels indicate the presence of some low-level non-Gaussianity. The standard noise estimation pipeline produces a positive bias in the power spectrum at high multipoles. Conclusions. In the case of a typical balloon-borne survey, the CR-induced non-Gaussianity will be marginally detectable in the membrane bolometer channels, but be negligible in the spider-web bolometer channels. In experiments with detector sensitivity better than 100 μK/ √ Hz, in an environment less favorable than the earth stratosphere, the CR-induced non-Gaussianity is likely to significantly affect the results.