I think the absolute record (excluding Euclid) belongs to

E. T. Whittaker G. H. Watson, A course of modern analysis.

According to the Jahrbuch database, the first edition was in 1915.
Moreover, this 1915 edition was an extended version of a 1902 book,
by Whittaker alone.

The last revision was in 1927.
The book is still in print, and widely used, not only by mathematicians
but by physicists and engineers.
Soon we will celebrate the centenary... It has 1056 citations on Mathscinet, by the way, and 8866 on the Google Scholar !

Perhaps this deserves a Guinnes book of records entry as a "textbook longest continuously in print".
And I suppose this is a record not only for math but for all sciences...
with the exception of Euclid and Ptolemy, of course:-)

If we include not only textbooks but research monographs there are plenty of other examples, even
older ones:

H. F. Baker, Abelian functions, was first published in 1897. Reprinted in 1995, and there is a new
Russian translation.

Just out of curiosity, look at its current citation rate in Mathscinet:-)

They also reprinted

H. Schubert, KalkĂĽl der abzĂ¤hlenden Geometrie, 1879, in 1979,

and again you can see from Mathscinet
that people are using this.

EDIT: A brief inspection of the most cited (and thus most used) books on Mathscinet shows that
a very large proportion of the most cited books are 30-40 years old.
Which is easy to explain, by the way. Thus on my opinion, such books do not qualify for this list
(unless we want to make it infinite).

EDIT2: Today I accidentally found that 3 of the 4 copies of

G. H. Watson, Treatise on the theory of Bessel functions (first edition, 1922)

are checked out from my university library.
Mathscinet shows 1157 citations for the last 2 editions.

Another question is old papers which are still highly sited. A typical life span of a paper is much
smaller than that of a book. In the list of 100 most cited papers in 2011, I found only two papers
published before 1950 (One by Shannon and another by Leray).

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