Answered April 22nd, 2019

Jim Brown attempts 37 rushes, setting a Browns team record that stood until Lee Suggs attempted 38 in 2004.
Who was the first Browns player with 37 rushing attempts in a game?
Interpreted as:
Who was the first Browns player with 37 or more rushing attempts in a game?
10/4/1959Brown12/26/2004Suggs12/27/2009Harrison373839
NAMEDATETMOPPATTRUSH YDSRUSH AVGRUSH TDREC YDSRECTRGREC AVGREC TD
Jim BrownJim Brown10/4/1959CLECLE@ CHC
@
371474.0222211.00
Lee SuggsLee Suggs12/26/2004CLECLE@ MIA
@
381433.804124.00
Jerome HarrisonJerome Harrison12/27/2009CLECLEvs. OAK
vs.
391483.810000
Results may be incomplete. StatMuse has complete game-level data for rush attempts going back to the 1960 season.