We extend non-Hermitian topological quantum walks on a Su-Schrieffer-Heeger (SSH) lattice [Phys. Rev. Lett. 102, 065703 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.065703] to the case of non-Markovian evolution. This non-Markovian model is established by coupling each unit cell in the SSH lattice to a reservoir formed by a quasicontinuum of levels. We find a topological transition in this model even in the case of non-Markovian evolution where the walker may visit the reservoir and return to the SSH lattice at a later time. The existence of a topological transition does, however, depend on the low-frequency properties of the reservoir, characterized by a spectral density J(ϵ)|ϵ|α. In particular, we find a robust topological transition for a sub-Ohmic (α<1) and Ohmic (α=1) reservoir, but no topological transition for a super-Ohmic (α>1) reservoir. This behavior is directly related to the well-known localization transition for the spin-boson model. We confirm the presence of non-Markovian dynamics by explicitly evaluating a measure of Markovianity for this model.