Mutations are the exception, and a favorable mutation would be even more of an exception. If a species is doing well in its environment, we would expect it to persevere, relatively unchanged. An *isolated* mutant variety could eventually develop into a new species, but the parent stock could continue to exist unless something comes along that wipes it out.
Over time, as a result of occasional isolated mutant populations, there can be a great variety of closely-related species. The mosquito, for example, is said to have 3,000 different species. Perhaps one of them is still the same as the original. This is not a problem for evolution.
The hangup is that for evolution to be true, to have produced millions of species living at present and putatively many times more throughout the ages, requires that species be able to change at a fairly large rate of speed. Not only that, since essentially (according to evolution) all species are descended from other species according to the evo so called 'tree of life', all species along the line must have been able to transform themselves for the new species to have ever arisen. So indeed stasis is a big problem for evolution. Evolution requires constant change.
Even bigger than the above problem though is the problem of the 'engine' of evolution - natural selection. Supposedly all species are 'molded' by natural selection as the environment constantly changes (as well as the competitor species). Now, if constant environment forces change in species and is what drives evolution, then it is clearly impossible for any species to have remained 'static' if evolution is true because at the minimum, the competitor species have been constantly changing. So yes, stasis is a strong disproof of evolution.