Aims:Our objective is to elucidate the physical process that causes the observed observed-minus-calculated (O-C) behavior in the M4.5/M4.5 binary CM Dra and to test for any evidence of a third body around the CM Dra system. Methods: New eclipse minimum timings of CM Dra were obtained between the years 2000 and 2007. The O-C times of the system are fitted against several functions, representing different physical origins of the timing variations. Results: Using our observational data in conjunction with published timings going back to 1977, a clear non-linearity in O-C times is apparent. An analysis using model-selection statistics gives about equal weight to a parabolic and to a sinusoidal fitting function. Attraction from a third body, either at large distance in a quasi-constant constellation across the years of observations or from a body on a shorter orbit generating periodicities in O-C times is the most likely source of the observed O-C times. The white dwarf GJ 630.1B, a proper motion companion of CM Dra, can however be rejected as the responsible third body. Also, no further evidence of the short-periodic planet candidate described by Deeg et al. (2000, A&A, 358, L5) is found, whereas other mechanisms, such as period changes from stellar winds or Applegate's mechanism can be rejected. Conclusions: A third body, being either a few-Jupiter-mass object with a period of 18.5 ± 4.5 years or an object in the mass range of 1.5 M_jup to 0.1 M&sun; with periods of hundreds to thousands of years is the most likely origin of the observed minimum timing behavior.