A new type of magnetically frustrated lattice is found in the layered fluoride NaBa2Mn3F11. A kagome-type array of regular triangles composed of Mn2+ ions (spin 5/2) deforms so as to generate the next-nearest-neighbor interaction J2 between three out of six spins in the hexagon of a normal kagome lattice, in addition to the nearest-neighbor interaction J1. As a function of |J2/J1|, this lattice can interconnect the kagome (J2 = 0) and the triangular (J2 = J1) lattices and thus is called the kagome-triangular (KT) lattice. Magnetic susceptibility and heat capacity measurements performed on a polycrystalline sample of NaBa2Mn3F11 show an intensive short-range antiferromagnetic correlation below 14 K probably due to the specific magnetic frustration of the KT lattice. In addition, a long-range order at 2.0 K is observed, which is significantly low compared with the antiferromagnetic Weiss temperature of 32.3 K. Theoretical considerations of Heisenberg spins in the KT lattice reveal unique non-coplanar magnetic orders in the case of ferromagnetic J1 and antiferromagnetic J2. NaBa2Mn3F11 may actually exist in this regime according to the results of our analysis based on classical Monte-Carlo simulation.