Yes, there is such an example.

This is nearly Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67, where the problem asks for a countably infinite $T_{2}$ example. In an editorial comment following John Cobb's solution in 107 #4 (April 2000), pp. 375-376, it is indicated that Prabir Roy's lattice space --- A countable connected Urysohn space with a dispersion point, Duke Mathematical Journal 33 #2 (1966), pp. 331-333 --- is an example that is a countable connected Urysohn space.

Yes, there is such an example.

This is nearly Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67, where the problem asks for a countably infinite $T_{2}$ example. In an editorial comment following John Cobb's solution in 107 #4 (April 2000), pp. 375-376, it is stated that Prabir Roy's lattice space --- A countable connected Urysohn space with a dispersion point, Duke Mathematical Journal 33 #2 (1966), pp. 331-333 --- is an example that is a countable connected Urysohn space. For what it's worth, my notes on this problem say that this can be verified by making use of the analog in Roy's space of observations (a) and (b) at the beginning of the proof of the theorem at the bottom of p. 375.

This is Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67 and whose published solution is by John Cobb in 107 #4 (April 2000), pp. 375-376.

Yes, there is such an example.

This is nearly Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67, where the problem asks for a countably infinite $T_{2}$ example. In an editorial comment following John Cobb's solution in 107 #4 (April 2000), pp. 375-376, it is indicated that Prabir Roy's lattice space --- A countable connected Urysohn space with a dispersion point, Duke Mathematical Journal 33 #2 (1966), pp. 331-333 --- is an example that is a countable connected Urysohn space.