Wrapping ETH into WETH via ethers -- WETH coins on my balance won't appear

I'm attempting to wrap ETH on Goerli into WETH by calling deposit function using ethers. Even though several TX already have been confirmed and successfully mined, nothing will appear on my WETH balance at goerli.etherscan.io in the dropdown Token Holdings. No is there WETH there at all, whereas there're other tokens

  async function myWrap(prKey) {
    let wallet = new ethers.Wallet(prKey, provider);
    const contract = new ethers.Contract(WETH_CONTRACT_ADDRESS, WETH_ABI, wallet);
    const txSigner = contract.connect(wallet);

    const balance = await provider.getBalance(wallet.address);
    const gasPrice = await provider.getGasPrice();
    const balanceMinusGas = balance.sub(gasPrice.mul(ETH_GAS_LIMIT));

    let tx = txSigner.deposit({value: balanceMinusGas});
    return tx;
  }

What's the matter?

Moreover, at https://goerli.etherscan.io/address/0xC02aaA39b223FE8D0A0e5C4F27eAD9083C756Cc2 my TX-s are present.

update1

Additionally, I then have tried to call

let amount = ethers.utils.parseEther(<sum of all previous ETH-s transfers>);
contract.transfer(wallet.address, amount);

yet, to no avail

update 2

I'd forgotten to mention that it was on Goerli. I've now switched to 0xB4FBF271143F4FBf7B91A5ded31805e42b2208d6 which is the correct address of WETH.

However, calling contract.deposit() still haven't gotten me any WETH. Although, again, all the calls have been successfully mined.

Can you share the exact tx link please?

You should add await in several places:

  1. let tx = await txSigner.deposit({value: balanceMinusGas});
  2. await contract.transfer(wallet.address, amount);

For 1, you can instead await when you call the function (or not at all, if calling this function is the very last action in your script).

For 2, it's not really clear where you're executing it, but generally speaking - same guidelines as 1.

No need in await there.

What's my enquiry?

...........