We calculate the rates for the charged lepton flavour violating decays l_i ->
l_j gamma, tau -> l pi, tau -> l eta, tau -> l eta', mu^- -> e^- e^+ e^-, the
six three body leptonic decays tau^- -> l_i^- l_j^+ l_k^- and the rate for mu-e
conversion in nuclei in the Littlest Higgs model with T-Parity (LHT). We also
calculate the rates for K_{L,S} -> mu e, K_{L,S} -> pi^0 mu e and B_{d,s} ->
l_i l_j. We find that the relative effects of mirror leptons in these
transitions are by many orders of magnitude larger than analogous mirror quark
effects in rare K and B decays analyzed recently. In particular, in order to
suppress the mu -> e gamma and mu^- -> e^- e^+ e^- decay rates and the mu-e
conversion rate below the experimental upper bounds, the relevant mixing matrix
in the mirror lepton sector V_Hl must be rather hierarchical, unless the
spectrum of mirror leptons is quasi-degenerate. We find that the pattern of the
LFV branching ratios in the LHT model differs significantly from the one
encountered in the MSSM, allowing in a transparent manner to distinguish these
two models with the help of LFV processes. We also calculate (g-2)_mu and find
the new contributions to a_mu below 10^{-10} and consequently negligible. We
compare our results with those present in the literature.Comment: 57 pages, 19 figures, 3 tables. Note added on the omitted term in
Z-penguin contribution. For an updated analysis see arXiv:0906.545
