Is the bottom of a pyramid one of the faces of a pyramid?

No, the bottom of a pyramid is the "Base" the sides of a pyramid are the "faces".

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face.

https://en.wikipedia.org/wiki/Pyramid_(geometry)

According to a by Euler: Faces+ Vertices - Edges = 2

Consider a triangular pyramid. That is a pyramid with a triangular base . It has 4 vertices. It has 6 edges. You now have : Faces +4 -6 =2 . The only way that can be true is if the pyramid has 4 faces. Therefore , the "base" is a face.

https://plus.maths.org/content/eulers-polyhedron-formula
https://www.mathsisfun.com/geometry/eulers-formula.html

Things can have more than one name, so because something is a base, doesn't mean it can't also be called a face.
Here is the mathematical definition of "face":

In solid geometry, a face is a flat surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.

Clearly all the flat surfaces of a triangular pyramid are faces, including the base, by this definition.

https://www.bing.com/search?q=mathematical+definition+of+face&qs=n&form=QBRE&sp=-1&pq=mathematical+definition+of+face&sc=4-31&sk=&cvid=063999ADD6804C59B183EBCEA1AEF0A1

As I read https://en.wikipedia.org/wiki/Pyramid_(geometry) the base is a face. In the infobox (at top right of article) it is stated that the faces are n triangles and 1 n-gon (a polygon or multiple-sided figure, n being the munber of sides). In the case of a tetrahedron, all the faces are triangles, so there are n=3 triangles + 1 triangle = 4 faces. If the base is square, you have n=4 triangles + 1 square = 5 faces. The base is simply a special case of face, being not necessarily a triangle like all the other faces. In the case of the Egyptian funereal pyramids, the base is not visible, and indeed may be virtual as the builders sometimes built over rocky outcrops to save on transporting filling. In those cases, the structure may technically not be a pyramid as the 'base' as built is not a plane surface. It is usual to treat it as though it were and no-one is likely to lift one up and move it. https://en.wikipedia.org/wiki/Egyptian_pyramid_construction_techniques If the sides are not triangles, it isn't a pyramid mathematically, but in terms of temple or tomb construction, a pyramid counts as a pyramid if the top would be a smaller pyramid which has been left off to give a working area for sacrificing, praying rather closer to the gods than street level, or even keeping a watch either on your own people or that nasty tribe from over there.... (BTW the base is usually at the bottom, but there is nothing to stop anyone building one with the square bit above ground. It'll look a bit silly, and probably won't last as long, that's all. And the local planning department probably wouldn't like it either.)