Yes; there have been several (although, to my knowledge, none in the most simple sense where seven binary bits are treated strictly as as a base-7 system of Peano-like numbers). Instead, they are systems in which at least one (typically, two or three) carry are treated as separate state-modification bits.
The most oldest/most simple example (although it may not meet the definition of a Turing-complete computer) is the ancient 5/2 abacus.
More recent examples generally are cases where some form of binary-coded decimal is used, particularly those that use Chen-Ho encoding (which fit a better conception of the system being "7-bit", as Boolean logic/operations can still be (relatively) easily applied, as opposed to more packed (or packed/padded) 7-bit numbers, which require a variable number of instructions to ascertain certain binary/two's complement values.
Of these, the "two of seven" approach is most common. Examples: the IBM 650, the FACOM 128, and the "IBM 370 compatibility feature" (hardware emulation) built in to the IBM 7070/7074.