Befunge-98 (PyFunge), 120 bytes
Try it online!
This is borderline in terms of the timelimit. 10,000 digits take around 11 seconds on my laptop, but I'm sure there must be a "reasonable" PC that could do it faster than that.
However, if you're trying it out on TIO, note that it won't return anything until it hits the 60 second time limit, since the algorithm is designed to keep going forever. By that time you'll have a lot more than 10,000 digits though.
I'm using the Jeremy Gibbons spigot algorithm, which I think is the same as most other answers here. However, note that this relies on the interpreter having arbitrary precision memory cells, and the only implementation I'm aware of that supports that is PyFunge.
cf*10p Initialise r to 180.
'<20p Initialise t to 60.
11 Initialise i and q on the stack to 1.
> Start of the main loop.
00p Save the current value of q in memory.
1+:30p Increment i and save a copy in memory.
:::*+39**6+ Calculate u = 27*(i*i+i)+6.
: Make a duplicate, since we'll need two copies later.
30g39**c-00g*10gv Calculate y = (q*(27*i-12)+5*r)/(5*t).
,+*86: Convert y to a character so we can output it.
*a*-*g02\+g01*g00-2*5g03 Calculate r = 10*u*(q*(i*5-2)+r-y*t)
p01 Save the updated r.
*g02 Calculate t = t*u
p02 Save the updated t.
>:2*1-*00g*a* Calculate q = 10*q*i*(i*2-1).
^ Return to the start of the main loop.