The transverse spin transfer from polarized protons to Lambda and (Lambda) over bar hyperons is expected to provide sensitivity to the transversity distribution of the nucleon and to the transversely polarized fragmentation functions. We report the first measurement of the transverse spin transfer to Lambda and (Lambda) over bar along the polarization direction of the fragmenting quark, D-TT, in transversely polarized proton-proton collisions at root s = 200 GeV with the STAR detector at RHIC. The data correspond to an integrated luminosity of 18 pb(-1) and cover the pseudorapidity range vertical bar eta vertical bar < 1.2 and transverse momenta p(T) up to 8 GeV/c. The dependence on p(T) and eta are presented. The D-TT results are found to be comparable with a model prediction and are also consistent with zero within uncertainties.