In the case that the pursuers have to actually catch the fugitive, this was answered in the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilse.

They actually prove that for any $\epsilon >0$, a fugitive running at speed at most $1+\epsilon$, can always escape countably infinitely many pursuers running at speed at most $1$. This holds for any starting position (provided no pursuer starts at the same point as the fugitive). The pursuers are even allowed to follow any pursuit strategy they like.

Note that in their version the fugitive and pursuers are allowed to slow down, but as pointed out by Alessandro Della Corte, this can be simulated by constant speed strategies.

In the case that the pursuers have to actually catch the fugitive, this was answered in the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilse.

They prove the following much more general theorem showing that the fugitive can always escape.

Theorem. For any $\epsilon >0$, a fugitive running at speed at most $1+\epsilon$, can always escape countably infinitely many pursuers running at speed at most $1$.

This holds for any starting position (provided no pursuer starts at the same point as the fugitive). The pursuers are even allowed to follow any pursuit strategy they like.

Note that in their version the fugitive and pursuers are allowed to slow down, but as pointed out by Alessandro Della Corte, this can be simulated by constant speed strategies.

Yes, the fugitive can always escape. See the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilse. They actually prove that for any $\epsilon >0$, a fugitive running at speed at most $1+\epsilon$, can always escape countably infinitely many pursuers running at speed at most $1$. This holds for any starting position (provided no pursuer starts at the same point as the fugitive). The pursuers are even allowed to follow any pursuit strategy they like.

Note that in their version the fugitive and pursuers are allowed to slow down, but as pointed out by Alessandro Della Corte, this can be simulated by a constant speed fugitive.

In the case that the pursuers have to actually catch the fugitive, this was answered in the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilse.

They actually prove that for any $\epsilon >0$, a fugitive running at speed at most $1+\epsilon$, can always escape countably infinitely many pursuers running at speed at most $1$. This holds for any starting position (provided no pursuer starts at the same point as the fugitive). The pursuers are even allowed to follow any pursuit strategy they like.

Note that in their version the fugitive and pursuers are allowed to slow down, but as pointed out by Alessandro Della Corte, this can be simulated by constant speed strategies.