Short answer is: yes, but not exactly the same as ARPANET

Experimental anti-missile defense system (called "System-A") has first prototypes of networked computers - in 1958. Data from Kazakhstan test firing range were sent to Moscow computing facility.

By June 1961 anti-missile defense system A-35 was in work. It included multiple radar stations, each with a mainframe computer. A state-wide network to connect all of them was planned too.

Deployment started in 1962. The progress was slow (as network was just a small part of quite complex radar-rocket system), but by 1977 all sites were operational.

Several differences vs ARPANET:

• A-35 network was never opened to civilian use.

• Other networks - in 80s X.25-based networks were all the rage. For example АКАДЕМСЕТЬ = AKADEMSET' was deployed to connect scientists in all Warsaw Pact countries.

• OGAS = ОГАС - which would be closest USSR analogue to Internet (as open, general-purpose network for enterprises and personal use) - was envisioned in 1959, but project was eventually killed. Some elements were prototyped.

Sources

1. (russian) Computer 5E92B : eternal soul of "Aldan" = Вычислительная машина 5Э92б: Бессмертная душа "Алдана". Skip first several paragraphs - it contains musing about popular sci-fi book of Strugatsky brothers about "Aldan-3" (fictional computer). A lot of implementation details and photos.

2. (russian) ABM A-35

3. (russian) Diagram of connections and data flows of "Aldan"

4. (russian) А. И. Берг, А. И. Китов, А. А. Ляпунов. О возможностях автоматизации управления народным хозяйством . Article by Colonel Kitov, acad. Lyapunov, acad. Berg with OGAS proposal, published in 1961, first presented in 1959 in seminar.

Just to add to what Acritas said, this article is pretty good, I'll summarise in the morning.