The parts added as edit to the original answer are in italics.
There is a misunderstanding about the conditions of a reversible process. Quasistatic is a necessary condition to ensure the well-definiteness of the thermodynamic state of the system. However, it is not sufficient. Even a quasistatic process becomes irreversible in the presence of dissipation (aka production of entropy).
A quasistatic process is a process slow enough (with respect to the relevant relaxation times of the system) to guarantee that, at each time, the system is as closest as possible to a thermodynamic equilibrium state. Missing such a condition, it would not be possible to characterize the process through well-definite thermodynamic state functions. The few-variable description of thermodynamics should be abandoned in favor of locally evolving dynamics described by many degrees of freedom. Think, for example, of the case of a fluid, where a non-quasi-static process would require the full set of local hydrodynamical variables to describe phenomena like sound waves, shock waves, and so on.
In addition to the need for a local field description, non-quasi-static phenomena usually involve internal mechanisms generating entropy in the system. Therefore, a quasistatic process is necessary for reversibility.
The opposite is not true: not every irreversible process corresponds to non-quasi-static processes. Actually, we may have quasi-static irreversible processes as soon as some additional mechanisms of production of entropy are present. One such case is the presence of dissipation.
Looking at this issue from a microscopic way, dissipation implies a transfer of energy into microscopic motion. Whether such a process is slow or fast doesn't matter. It will affect only the power dissipated (energy per time unit).
Whether such energy transferred to microscopic degrees of freedom should be called heat or otherwise is, in a way, a matter of conventions without effects on the phenomena.
Such a statement may look odd, but it is present in the literature (see, e.g., Mungan, C. E. (2007). Thermodynamics of a block sliding across a frictional surface. The Physics Teacher, 45(5), 288-291 where the author concludes by writing "while Q and W have important
roles in the introductory teaching of the reversible
thermodynamics of simple systems such as ideal gases,
students should eventually be brought to realize that it
is not always convenient nor necessary to categorize all
channels of energy transfer as either “heat” or “work.” )
As an illustration, I can notice that if we define heat every variation of internal energy of the system not amenable to macroscopic work or matter transfer (i.e., the definition used by Caratheodory, Planck, Born, and others), the answer to the original question still depends on what we want to consider as the thermodynamic system. If the surface where friction is present is not considered part of the thermodynamic system, friction at the wall should be accounted as heat. If it is part of the system, we must evaluate friction forces' work.
However, what is physically meaningful is the variation of internal energy of the system due to the interaction with its environment.