I'm not a biology person so please forgive me if this question is formulated badly :)
I'm curious. Are there any species that has an ancestor species that is still alive today?
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First, you probably want to have a look at this answer who offers an introduction to phylogeny.
There are few subjects that yield to as many semantic issues than the definition of species and the definition of life. Your question is at the border of the species definition. Therefore the question is not so much about Biology than about Philosophy, so let me advertise: The following answer is nothing else than a semantic discussion (philosophy) and has no relevance of our understanding of evolutionary processes. Note however that semantic discussions might be very helpful to evolutionary biologists as well.
Dogs and wolves
Commonly, in sexual organisms, species are defined by the concept of reproductive isolation. To address @WYSIWYG's comment, wolves can breed with dog. The offspring is called a wolfdog and it suffers from little inbreeding depression.
Is the MRCA contained in the species of one descendent
But I'll use the wolf example anyway and make some fake assumptions. There are probably better examples I could pick though. Let's assume that wolves and dogs can't interbreed, that they are therefore separate species. Let's assume as well that somehow we manage to recreate the Most Recent Common Ancestor (MRCA) of wolves and dogs (which has some issue due to parental effect incl. culture and epigenetic) and this MRCA can breed with wolves and but not with dogs, then we could eventually say that the MRCA and the current wolf (there are actually 3 species of wolves that are probably not real species) are the same species. One could then say that dogs descend from the wolf.
Feeling that we talk about improvement over another species
I think that much of the issue in saying that the dog descend from the wolf is that some people would understand things like "dog is more evolved than the wolf", which makes no sense. Both the current wolf and the current dog has evolved for exactly the same amount of time. Any living species on earth has evolved for exactly the same amount of time (almost 4 billion years).
Temporal constraint in the definition of species
Therefore, it is eventually possible to say that one node on a phylogenetic tree is of the same species that the previous and the next node. I think that in reaction to the misleading feeling that early evolutionary biologists and current non-biologists people have in hearing that one species descent from another one, we tend to make a temporal constraint to the species. A species is what it is only at the exact moment we talk about it. This prevents us to consider a MRCA as being of the same species of any descendent.
Note that the definition of species yield to various issues such as the case of ring species.
Same issue as above
Now, if we talk about asexual species. There is no clear definition of what is a species there. Of course if we were to use the definition based on reproductive isolation, then any asexual individual would be a new species. We therefore define a species when we see a group of phylogenetically related individuals that all share some feature that we find "interesting enough" (very arbitrary) to describe this group as a different species from the some other group of individuals. In asexuals, you may see one lineage that accumulate 10 mutations, while the other lineage accumulates 1000 mutations (this other lineage may eventually live in a different environment than the MRCA of the two lineages). Therefore, you get to the same issue when thinking about asexuals than about sexual organisms. If one lineage ressemble more their MRCA than the other lineage, then is their MRCA part of the species of one lineage?
Is a species a thing?
Well... the concept of species is useful in the sense that it is often not ambiguous to classify one individual into one species or into another. The concept of species allows us to better communicate.. so yes I guess it is useful.
The concept of species existed way before we had any understanding of natural selection, mutations, genetic drift and other basic concepts in evolutionary biology. Aristotle used the concept of species (he also thought without having any good evidence that life evolves). This concept of species has been very well conserved because of the static view we had on what a species is before Darwin. Now that we have a better understanding of evolutionary biology, ecology and reproductive biology we discover that the concept of species is not a natural category but is a man-made unclear category.
Other discussion on the subject
In this post I made an overview of the issue related to the concept of species. I did not go into the details but tried to list important concepts.
You will find many websites that address the concept of species in philosophy. I particularly like discussion in the concern of how current bad semantic may yield to bias in the way we think nature and may yield to bad decision in applied science. Here is a podcast (in french) on the origin and consequences of the concept of species.