In this excellent answer to How have space suits dissipated the heat removed from astronauts? I learned that space suits from the Apollo missions and even when used in space walks from the ISS today use water sublimation to remove both body heat and other heat generated inside.
In the NASA document Portable Life Support System (Hamilton Standard, undated) - one of the sources linked in that answer - it says:
This subsystem supplies 11.8 pounds of expendable water, stored in a rubber bladder reservoir, to the heat-rejecting porous-plate sublimator. Of this expendable feedwater, 8.5 pounds is stored in the main reservoir; an auxiliary tank holds the remaining 3.3 pounds. Suit pressure against the bladder forces water into passages between the sublimator’s heat transport fluid passages and its metal plates, which are exposed to space vacuum. The ice layer formed on the porous plates during sublimation prevents the slightly pressurized water from flowing through the metal pores.
The concept of having ten pounds of extra, or "expendable water" to lose every time you go outside to the Martian surface seems a little out of reach, at least for early missions or adventure.
The Enthalpy of Sublimation of water is about 51 kJ / mole , or about 2.8 million J/kg or 2700 BTU/kg. Suit electronics and mechanics will be much more efficient and therefore cooler than they were in 1970. For a nominal 1000 BTU/hour, assuming 70% overall thermodynamic efficiency of the sublimation system (just a guess), that means expending water to the Martian atmosphere at 0.5 kg per hour.
Is this actually what's probably going to happen if/when there are suited people walking around on Mars? Maybe there will be modular ice packs sitting around on the surface that one could swap out regularly, with a water-lossy sublimator as backup only?
Edit: Both human bodies, and associated electronics and electromechanics generate heat and will cease to operate fairly quickly if you don't get the heat OUT of the suit or packaging. This question is strictly about how to get rid of heat while on the surface. I'd like an answer that is at least roughly defensible quantitatively.