We give a proof of the irrationality of the p-adic zeta-values ζp​(k)
for p=2,3 and k=2,3. Such results were recently obtained by F.Calegari as
an application of overconvergent p-adic modular forms. In this paper we
present an approach using classical continued fractions discovered by
Stieltjes. In addition we show irrationality of some other p-adic L-series
values, and values of the p-adic Hurwitz zeta-function