The 19-year-old Alpine junior driver is second in the standings heading to Silverstone this weekend, albeit sitting 37 points behind unbeaten points leader Chadwick.
Pulling made four guest outings in W Series last year and earned an automatic place on the grid for this season by finishing in the top eight with a pole and a podium at the Circuit of the Americas to her name.
The Briton, who finished eighth on her debut on her series debut at Silverstone last year, was second to Chadwick last time out at Barcelona and gave the double series champion her closest race of the year by finishing just 0.526s in arrears.
However, Pulling said she isn't feeling any additional pressure compared to her first series outing and is "past the point of pinching myself now".
"I've matured as a driver and changed a lot, and I showed that at the last race in Barcelona," she said.
"I want to keep coming away from race weekends feeling like I've done as well as I can.
"When I'm at my best mentally, I perform at my best on-track and the results come, just like it did in Barcelona where I drove a composed race.
"I showed Jamie [Chadwick] that I'm not going anywhere, and I want to keep the pressure on her and not give her a break until the season ends.
"People might say there's more pressure on me at Silverstone this year because of where I am in the championship, but I don't feel that.
"Last year, I told myself I needed to perform to get a spot for 2022, but now I can focus purely on the job in hand.
"That's what I mean about maturing as a driver. I concentrate on what I need to do on-track as opposed to thinking about things I can't really control."
Last year's race was won by Pulling's mentor Alice Powell, who passed Fabienne Wohlwend late on as Chadwick finished third.
The 24-year-old said she has "some unfinished business after last year, when I didn't have my strongest weekend".
"Never in my career have I had such a good start to a season, and I'm pleasantly surprised with how well it's going," reflected Chadwick.
"I have momentum and I feel confident, but I know things can change quickly and I'm under no illusions because there are still lots of races to come."
Chadwick added that she's "not thinking about the points gap or the championship standings" and is focused on "winning as many races as I can for the rest of the year".
"Right now, I have lots of confidence in the car and feel like I can deliver my best every weekend," she said.
