To answer your question properly I need to understand what you already know about DNA, genes, genomes, mutations, and speciation. I will assume you have some basic knowledge (let me know if I assume too much).

Your question is really about the interactions between the rates of mutational change and the drivers of speciation. The short answer is "yes, random mutation together with other well described molecular level processes and selection is enough to account for the diversity of life."

Let us first consider mutations. A mutation in a gene will generate a new form of the gene, known as an allele. Mutations in non-gene regions of the genome will generate sequence differences, known generically as polymorphisms.

As demonstrated directly through in a comparative study of vinegar fly Drosophila species, random mutations in regions of the genome that did not originally contain genes can generate new genes that improve reproductive success, and so are favored by natural selection (see Zhao et al., 2014).

Genome level changes, ranging from capturing genes through horizontal gene transfer (see this review), DNA duplications, deletions, and rearrangements, particularly those associated with sexual reproduction, also play a critical role; the genome is dynamic (see Bergthorsson et al., 2007).

We can measure mutation rates in various ways. For example, as part of the long term evolution study using the bacterium E. coli, Barrick et al., (2009) and Wielgoss et al (2011), compared cells at the start and after various numbers of generations.

At 20,000 generations (the outer ring in this figure →) they found 45 mutations in the population, 29 were changes in a particular base pair, known as a single nucleotide polymorphism or SNP, and 16 insertions, deletions, or inversions. They estimated that the mutation rate was approximately 9×10⁻¹¹ per base-pair per generation.

They also followed this (and other strains) for another 20,000 generations, at which point there were 627 SNPs and 26 insertions, deletions, and inversions. Moreover they found two surprising results, first they found mutations that increased the mutation rate by about 150 fold, and then later independent mutations that reduced the high mutation rate by about 50%.

This illustrates the fact that the rate at which mutations occur is itself a selectable trait (part A - below).

As first noted by Muller in the 1930's (see Henson et al., 2012), the effects of a mutation on a gene can range from abolishing or reducing its activity to increasing its activity, to producing a new activity.

The impact of mutations on reproductive success, which is what matters evolutionarily, can range from the highly negative (such as the inability to make proteins) to the highly positive (such as the ability to survive in the presence of an antibiotic.) In practice, most mutations are either neutral in their effects or weakly negative.

Where selection effects are weak, whether a particular mutant allele increases or decreases in frequency within the population will also be influenced by genetic drift, that is, the stochastic factors that influence reproductive success.

The other version of the genes (alleles) in a genome influence the effects of a particular mutation. This was clearly illustrated in the observation that the evolution of the ability to metabolize citrate was made possible by mutations (potentiation events) that originally had no dramatic selective benefit in themselves but made the evolution of citrate metabolism by E. coli possible in later generations. (see Blount et al., 2012)(part B - above).

Similarly, the origin of the mutator phenotype (around generation 26,000) increased the ability of cells to develop resistance to various antibiotics.

Speciation: This raises the second point. We need to consider what drives speciation. Typically populations of a particular organism are spread over various environmental conditions.

Evolutionary pressures will favor organisms that can avoid competition by specializing in order to occupy a particular (and currently unoccupied) ecological niche or who can out compete members of other species for a currently occupied niche (which is harder). For example, the evolution of the ability to metabolize citrate in E. coli opens up a potentially new ecological niche.

A key feature is that as populations begin to specialize to a particular niche. Interbreeding between populations, if it occurs at all, can be strongly selected against (a process known as reproductive isolation).

Over time, this leads to two distinct species emerging from one ancestral population.

It is this process that has led to the millions of species that currently exist and the many millions more that existed in the past and are now extinct. All current evidence indicates that all current organisms are descended from a common ancestral, probably unicellular, organism.

All of which is to say that based on empirical measures of mutation rates and genomic dynamics, the time available for evolutionary change, (millions of years, the equivalent to many thousands of generations for each speciation event), there appears to be no serious obstacle to the premise that random mutation together with selection driven adaptation to specific ecological niches is sufficient to drive the patterns of speciation we see around us and in the fossil record.