We investigate in detail the LiTE of the most probable binary candidate TU UMa. We used a new code applied on previously available and newly determined maxima timings to confirm binarity and refine parameters of the orbit of the RRab component in the binary system. The binary hypothesis is also tested using RV measurements. We used new approach to determine brightness maxima timings based on template fitting. This can also be used on sparse or scattered data. This approach was successfully applied on measurements from different sources. To determine the orbital parameters of the double star TU UMa, we developed a new code to analyse LiTE that also includes secular variation in the pulsation period. Its usability was successfully tested on CL Aur, an eclipsing binary with mass-transfer in a triple system that shows similar changes in the O-C diagram. Since orbital motion would cause systematic shifts in mean RVs (dominated by pulsations), we computed and compared our model with centre-of-mass velocities. They were determined using high-quality templates of RV curves of RRab stars. Maxima timings adopted from the GEOS database (168) together with those newly determined from sky surveys and new measurements (85) were used to construct an O-C diagram spanning almost five proposed orbital cycles. This data set is three times larger than data sets used by previous authors. Modelling of the O-C dependence resulted in 23.3-year orbital period, which translates into a minimum mass of the second component of about 0.33 M_s. Secular changes in the pulsation period of TU UMa over the whole O-C diagram were satisfactorily approximated by a parabolic trend with a rate of -2.2 ms/yr. To confirm binarity, we used RV measurements from nine independent sources. Although our results are convincing, additional long-term monitoring is necessary to unambiguously confirm the binarity of TU UMa.